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COMPRESSED AIR 
THEORY AND COMPUTATIONS 



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Published by the 

'raw- Hill Book. Company 



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COMPRESSED AIR 



THEORY AND COMPUTATIONS 



BY 

ELMO G. HARRIS, C.E. 

PROFESSOR OF CIVIL ENGINEERING, MISSOURI SCHOOL OF MINES, 

IN CHARGE OF COMPRESSED AIR AND HYDRAULICS; 

MEMBER OF AMERICAN SOCIETY OF 

CIVIL ENGINEERS 



c • 

> • • 



McGRAW-HILL BOOK COMPANY 

239 WEST 39TH STREET, NEW YORK 

6 BOUVERIE STREET, LONDON, E.C. 

1910 






Copyright, 1910, 

BY THE 

McGRAW-HILL BOOK COMPANY 



Stanbope ipress 

F. H. GILSON COMPANY 
BOSTON, U.S.A. 






/ 



■>CLA271420 



PREFACE 



This volume is designed to present the mathematical 
treatment of the problems in the production and applica- 
tion of compressed air. 

It is the author's opinion that prerequisite to a successful 
study of compressed air is a thorough training in mathe- 
matics, including calculus, and the mathematical sciences, 
such as physics, mechanics, hydraulics and thermodynamics. 

Therefore no attempt has been made to adapt this volume 
to the use of the self-made mechanic except in the insertion 
of more complete tables than usually accompany such work. 
Many phases of the subject are elusive and difficult to see 
clearly even by the thoroughly trained; and serious blunders 
are liable to occur when an installation is designed by one 
not well versed in the technicalities of the subject. 

As one advocating the increased application of compressed 
air and the more efficient use where at present applied, the 
author has prepared this volume for college-bred men, believ- 
ing that such only, and only the best of such, should be 
entrusted with the designing of compressed-air installations. 

The author claims originality in the matter in, and the use 
of, Tables I, II, III, V, VI, VII and IX, in the chapter on 
friction in air pipes and in the chapter on the air-lift pump. 

Special effort has been made to give examples of a practical 
nature illustrating some important points in the use of air 
or bringing out some principles or facts not usually appre- 
ciated. 

Acknowledgment is herewith made to Mr. E. P. Seaver 
for tables of Common Logarithms of Numbers taken from 
his Handbook. 



CONTENTS 

Page 

Symbols and Formulas ix, xi 

CHAPTER I. 

Art. 1. Formulas for Work. — Temperature Constant. . . 1 

Art. 2. Formula for Work. — Temperature Varying. .. . 3 

Art. 3. Formula for Work. — Incomplete Expansion ... 7 

Art. 4. Effect of Clearance. — In Compression 8 

Art. 5. Effect of Clearance and Compression in Expan- 
sion Engines 9 

Art. 6. Effect of Heating Air as it Enters Cylinders .... 11 
Art. 7. Change of Temperature in Compression or Ex- 
pansion 12 

Art. 8. Density at Given Temperature and Pressure ... . 13 

Art. 9. Cooling Water Required 14 

Art. 10. Reheating and Cooling 14 

Art. 11. Compounding 16 

Art. 12. Proportions for Compounding 19 

Art. 13. Work in Compound Compression 20 

Art. 14. Work under Variable Intake Pressure 21 

Art. 15. Exhaust Pumps 22 

Art. 16. Efficiency when Air is Used without Expansion. . 24 

Art. 17. Variation of Free Air Pressure with Altitude. . . 25 

CHAPTER II. Measurement of Air. 

Art. 18. General Discussion 27 

Art. 19. Apparatus for Measuring Air by Orifice 28 

Art. 20. Formula for Standard Orifice under Low Pressure 29 

Art. 21. Air Measurement in Tanks 30 

CHAPTER III. Friction in Air Pipes. 

Art. 22. General Discussion 33 

Art. 23. Friction Formula assuming Density and Tem- 
perature Constant 33 

Art. 24. Theoretically Correct Friction Formula 36 

Art. 25. Efficiency of Power Transmission by Compressed 

Air 39 

CHAPTER IV. Hydraulic and Centrifugal Air Compres- 
sors. 

Art. 26. Displacement Type of Air Compressor 41 

Art. 27. Entanglement Type of Air Compressor 42 

Art. 28. Centrifugal Type of Air Compressor 44 

vii 



Vlll CONTENTS 

Page 
CHAPTER V. Special Applications of Compressed Aib. 

Art. 29. The Return- Air System — In General. . . . 45 

Art. 30. The Return-Air Pumping System 46 

Art. 31. The Simple Displacement Pump 48 

CHAPTER VI. The Air-Lift Pump. 

Art. 32. General Discussion 49 

Art. 33. Theory of the Air-Lift Pump 49 

Art. 34. Design of Air-Lift Pumps 52 

Art. 35. The Air Lift as a Dredge Pump 57 

Art. 36. Testing Wells with the Air Lift 57 

Art. 37. Data on Operating Air Lifts 58 

CHAPTER VII. Examples and Exercises 

Art. 38. ... , 60 

Plates and Tables 71 

Appendix A. Drill Capacity Tables 96 

Appendix B. Data on Friction in Air Pipes 100 



SYMBOLS 



For ready reference most of the symbols used in the text are assembled 
and denned here. , 

V = intensity of pressure (absolute), usually in pounds per square 

foot. Compressed-air formulas are much simplified by 
Using pressures measured from the absolute zero. Hence 
where ordinary gage pressures are given, p = gage pres- 
sure + atmospheric pressure. In the majority of formulas 
p must be in pounds per square foot, while gage pressures are 
given in pounds per Square inch. Then p = (gage pressure + 
atmospheric pressure in pounds per square inch) X 144, 

V = volume — usually in Cubic feet. 

Where sub-a is used, thus pay Va-, the symbol refers to free 
air conditions. 

,, j. . . higher pressure 

f = ratio of compression or expansion = «-^ 

lower pressure 

The lower pressure is not necessarily that of the atmos- 
phere. 

t — absolute temperature = Temp. F. -f- 460.6. 

n = an empirical exponent varying from 1 to 1.41. 

log e = hyperbolic logarithm = (common log.) X 2.306. 

W = work ~*= usually in foot-pounds per second. 

Q = weight of air passed in unit time. 

w = weight of a cubic unit of air. 

Other symbols are explained where used. 



is 



FORMULAS 



For convenience of reference the principal formulas appearing in the 
text are collected here with the article and page where demonstration 
and complete explanation can be found. 



No. Formula Ar 

1. W = pv log e r 

la. W = 53.17 t log e r for one pound 

lb. W = (122.61 logior) t for one pound 

2. W = 63737 log 10 r for one pound at 60° F 

logic 63737 = 4.8043894. 

3 m = t l ;.... 2 

P2V2 t 2 

4. pv = 53.17 1 for one pound . 2 

5. p\V X n = p-zv 2 n , 2 



6. 
7. 

8. 
8b 



W = 
w = 
w = 



P2V2 — P1V1 



n - 1 
n 



+ P2V2 — P1V1 2 



(P2V2 — P1V1) 2 



n - 1 

[ II,— L -1 

r n - lj 



n-i 



n - 1 
W =-- 95190 (r - 29 - 1) for 1 lb. at 60° F., n = 1.41 



2 

8c. W = 138405 (r°- 2 -l) for lib. at 60° F.,n = 1.25. . . 2 
logio 95190 = 4.978606, log 138405 = 5.141141. 

_ / TO — 1 

w 



8d. 

9. 
10. 
10a. 

11. 



tjt P2V2 — P\V\ . c ,. , 

W = — j — + P2V2 — p a V\ for partial expansion . 

E v = l+c(l 

v 2 _ c + k 
Vi ~ c + 1 



r) volumetric efficiency 4 

5 



1 - 

\V2 



n-l 



11a. t. 



2 = tl l^y =tl (r) n 7 



Page 

2 
2 
2 

2 



= [~53.17 ^— ( r n _ i)j ^ for one p 0un d ...... 2 5 



10 
11 

13 
13 



XI 



xn FORMULAS 

No. Formula Art. Page 

12. vi — , Q = weight per cubic foot 8 13 

OO.l 4 t 

12a. w ** 2.708 [^ §+p ) = weight per cubic foot .... 8 13 
12b. d 2 — —y=i ; d 3 = — : diameters, stage compression.. 12 19 

( re — l v 
rf^~ -ljx 2; two-stage work 13 20 

( n — 1 v 
r 2 2n -ljx2; two-stage work 13 20 

7l> — 1 

/ n — i v 

13b. Tf ="_- p aVa \n tl -1JX 3; three-stage work. . . 13 21 

n — 1 

( re— _l \ 
r 3 3 rt - 1JX 3; three-stage work.. . 13 21 

74 — 1 

14. m = £r- 15 23 



n — 1 

Vo 




l0g V + v 
r - 1 



, 15. #=*^ — L . ,.. i 6 24 

r loge r 

16. p<*=* .4931 m{\- .0001 (F - 32)] 17 25 

17. log p a = 1.16866 - ^~ t 17 26 

18. Weight Q = c .1632 d 2 ^~p a ;p a in lbs. per sq. in., ... 20 29 
18a. Q - c .6299 d 2 V \ at sea level 20 30 

20. / = cj- % - • 23 35 

J d b r 

21. d dc l j^\ l 23 35 

24. logpf ^logpi-a-^^f-^Y 24 37 

Wad a \p x J 

25. 2?-!-^ 25 39 

logn 

26. E = J^ 26 42 

r — 1 

27. -g- - JL £ _i-_ 33 53 

Q 77.3 # logior 

28. s* = v«[l-|(l -^Yj 33 56 



COMPRESSED AIR 



CHAPTER I 
Formulas for Work 

Art. 1. Temperature Constant or Isothermal Conditions. 

From the laws of physics (Boyle's Law) we know that 
while the temperature remains unchanged the product pv 
remains constant for a fixed amount (weight) of air. Hence 
to determine the work done on or by air confined in a cylinder, 
or like conditions, when conditions are changed from pivi to 
P2V2 we can write 

PlVi = p x V x = P2V2, 

sub x indicating variable intermediate conditions. 




_ 1 



U 




Tfr 



E? 



Fig. 1. 



Whence p x = 2i-* and dW = p x Adl = p x dv x since Adl = dv; 

V x 

j 

A being the area of cylinder, therefore dW = piVi — , and 

Vx 



z COMPRESSED AIR 

work of compression or expansion between points B and C 
(Fig. 1) is the integral of this, or 

Jf*Vi (}p 
— = PlVi (log e Vx — log e v 2 ) 

= Pl^l l0g e - = Jp Y Vi log e — = P&1 l0g e T = p 2 V 2 log e T. 

v 2 Pi 

Note that this analysis is only for the work against the front 
of the piston while passing from B to C. To get the work 
done during the entire stroke of piston from B to D we must 
note that throughout the stroke (in case of ordinary compres- 
sion) air is entering behind the piston and following it up 
with pressure p\. Note also that after the piston reaches C 
(at which time valve / opens) the pressure in front is constant 
and = p2 for the remainder of the stroke. Hence the com- 
plete expression for work done by, or against, the piston is 

PxViloger - piVi + p 2 v 2 ; 
but since piVi = p 2 v 2 , the whole work done is 

W = PiVi loge T Or p 2 V 2 loge T. (1) 

Note that the operation may be reversed and the work 
be done by the air against the piston, as in a compressed-air 
engine, without in any way affecting the formula. 

Forestalling Art. 2, Eq. (4), we may substitute for pv in 
Eq. (1) its equivalent, 53.17 t, for one pound of air and get 
for one pound 

W = 53.17 ZX loge r. (la) 

This may be adopted for common logs by multiplying by 
2.3026. It then becomes 

W = (122.61 logio r) t, (lb) 

(log 122.61 = 2.0878852.) 

Note that in solving by logs the log of log r must be taken. 

Values of the parenthesis in Eq. (lb) are given in Table I 

For the special temperature of 60° F. (lb) becomes for one 

pound of air 

W = 63737 log 10 r, (2) 

log 63737 = 4.8043894. 



FORMULAS FOR WORK 3 

Example la. What will be the work in foot-pounds per 
stroke done by an air compressor displacing 2 cubic feet per 
stroke, compressing from p a = 14 lbs. per sq. inch to a gage 
pressure = 70 lbs.; compression isothermal, T = 60° F.? 

Solution (a): 

Inserting the specified numerals in Eq. (1) it becomes 

W = 144 X 14 X 2 X log e — H- 14 = 4032 X 1.79 = 7217. 

Solution (b): By Tables I and II. 

By Table II the weight of a cubic foot of air at 14 lbs. and 
60° is .07277, and .07277 X 2 = .14554. The absolute t is 
460 + 60 = 520, and r = 6.0. 

Then in Table I, column 11, opposite r = 6 we find 95.271, 

whence 

W = 95.271 X 520 X .14554 = 7208. 

The difference in the two results is due to dropping off the 
fraction in temperature. 

Art. 2. Temperature Varying. 

The conditions are said to be adiabatic when, during com- 
pression or expansion, no heat is allowed to enter in, or 
escape from, the air although the temperature in the body 
of confined air changes radically during the process. 

Physicists have proved that under adiabatic conditions 
the following relations hold: 

fcfc^fe ■ (3) 

P2V2 t 2 

and since for one pound of air at 32° F. pv = 26,214 and t = 

492.6, we get for one pound at any pressure, volume and 

temperature, 

pv = 53.17 t. (4) 

While formulas (3) and (4) are very important, they do not 
apply to the actual conditions under which compressed 
air is worked, for in practice we get neither isothermal nor 
adiabatic conditions but something intermediate. 

For such conditions physicists have discovered that the 
following holds nearly true: 

PiVi n = p x v x n = p 2 V2 n , (5) 



4 COMPRESSED AIR 

sub x indicating any intermediate stage and the exponent n 
varying between 1 and 1.41 according to the effectiveness 
of the cooling in case of compression or the heating in case of 
expansion. From this basic formula (5) the formulas for 
work must be derived. 

As in Art. (1) dW = p x dv x = piVi n — = pxVi 71 (y x ~ n ) dv x . 

v x n 
Therefore 

W'=vVi n j Vl v x - n dv x = PiV ^'lZn" ) = VlVin (~-^~^) ' 

Now since pxVi 11 X v 2 1 ~ n = p2V2 n X V2 l ~ n — p 2 ^and paV"" 
= piVi the expression becomes 

w , = P2V2 — P1V1 
n — 1 

which represents the work done in compression or expansion 
between B and C, Fig. 1. To this must be added the work 
of expulsion, p 2 v 2 , and from it must be subtracted the work 
done by the air entering behind the piston, piVi. Hence the 
whole net work done in one stroke is 

w = M^m + PiV2 _ PlVl (6) 

n — 1 

(P2V2 - P1V1). (7) 



n — 1 

Equation (7) is in convenient working form and may be used 
when the data are in pressures and volumes, but it is common 
to express the compression or expansion in terms of r. For 
such cases a convenient working formula is gotten as follows: 



From Eq. (5) p 2 v 2 - PlVl ?_? 

V2 



n-l 
n-1 



n 1 

Also r — ^ = — , therefore - = r n , 

Pi v 2 n v 2 



n-l 



n— 1 n— 1 



and — — 7 = r n » therefore p 2 v 2 = PiVtf n 

v 2 n ~ 1 

n r — 1 

and Eq. (7) becomes W = — — r pivA r n — 1 1. 



(8) 



FORMULAS FOR WORK 5 

The most common uses of equations (7) and (8) are when 
air is compressed from free air conditions, then pi and Vi be- 
come p a and v a . This case must be carefully distinguished 
from the case of incomplete expansion as presented in Art. 3. 

In perfectly adiabatic conditions n = 1.41, but in practice 
the compressor cylinders are water-jacketed and thereby 
part of the heat of compression is conducted away, so that 
n is less than 1.41. For such cases Church assumes n = 1.33 
and Unwin assumes n = 1.25. Undoubtedly the value 
varies with size and proportions of cylinders, details of 
water-jacketing, temperature of cooling water and speed of 
compressors. Hence precision in the value of n is practi- 
cable. Fortunately the work does not vary as much as n 
does. 

For one pound of air at initial temperature of 60° F. 
Eq. (8) gives in foot-pounds, 

When n = 1.41, W = 95,193 (r 029 - 1). (8b) 

When n = 1.25, W = 138,405 (r 02 - 1). (8c) 

Common log of 95,193 = 4.978606. 
Common log of 138,405 = 5.141141. 

The above special values will be found convenient for 
approximate computations. For compound compression 
see Art. 12. 

If in Eq. (8) we substitute for pv its value, 53.17 t, for 
one pound, we get 

[r^~ -l)\xt = kt, (8d) 

where k = - JL ~ X 53.17(r~"~ - l) . 

n — 1 

Table I gives values of k for n = 1.25 and n = 1.41 and 
for values of r up to 10, varying by one-tenth. The theoretic 
work in any case is K X Q X t, where Q is the number of 
pounds passed and t is the absolute initial temperature. 
Further explanation accompanies the table. 

The difference between isothermal and adiabatic compres- 
sion (and expansion) can be very clearly shown graphically 



w = 


lU-J 53 


k- 


n X53 



6 



COMPRESSED AIR 



as in Fig. 2. In this illustration the terminal points are 
correctly placed for a ratio of 5 for both the compression and 
expansion curve. 

f e d 






(V/ 


a - — ^ 




S, 


h! 





Fig. 2. 



Note that in the compression diagram (a), the area between 
the two curves aef represents the work lost in compres- 
sion due to heating, and the area between the two curves 
aeghb in (6) represents the work lost by cooling during 
expansion. The isothermal curve, ae, will be the same in 
the two cases. 

Such illustrations can be readily adapted to show the 
effect of reheating before expansion, cooling before compres- 
sion, heating during expansion, etc. 

Example 2a. What horse power will be required to com- 
press 1000 cubic feet of free air per minute from p a = 14.5 
to a gage pressure = 80, when n = 1.25 and initial tempera- 
ture = 50° F.? 

Solution. From Table II, interpolating between 40° and 
60° the weight of one cubic foot is .07686 and the weight of 
1000 is 76.86 -. The r from above data is 6.5. Then in 



FORMULAS FOR WORK 7 

Table I opposite r = 6.5 in column 9 we find .3658. Then 



Horse power = .3658 X 



76.86 
100 



X 510 = 143. 



The student should check this result by Eq. (8) or (8d) with- 
out the aid of the table. 

Art. 3. Incomplete Expansion. 

When compressed air is applied in an engine as a motive 
power its economical use requires that it be used expansively 
in a manner similar to the use of steam. But it is never 
practicable to expand the air down to the free air pressure, 
for two reasons : — first, the increase of volume in the cylin- 
ders would increase both cost and friction more than could 
be balanced by the increase in power; and second, unless 
some means of reheating be provided, a high ratio of expan- 
sion of compressed air will cause a freezing of the moisture 
in and about the ports. 

The ideal indicator diagram for incomplete expansion is 
shown in Fig. 3. In such diagrams it is convenient and 




Fig. 3. 

simplifies the demonstrations to let the horizontal length 
represent volumes. In any cylinder the volumes are pro- 
portional to the length. 

Air at pressure p 2 is admitted through that part of the 
stroke represented by v 2 — thence the air expands through 
the remainder of the stroke represented by vi, the pressure 
dropping to p\. At this point the exhaust port opens and 
the pressure drops to that of the free air. The dotted por- 
tion would be added to the diagram if the expansion should 
be carried down to free air pressure. 



8 COMPRESSED AIR 

To write a formula for the work done by the air in such a 
case we will refer to Eq. (6) and its derivation. In the case 
of simple compression or complete expansion it is correctly 
written 

w = *** - p.». + ViVl _ VaVaj 

n — 1 

which would give work in the case represented by Fig. 1 
when there is a change of temperature, but in such a case as 
is represented by Fig. 3 the equation must be modified thus : 

W = Mi^Pi + PiVi _ PaVl> (9) 

n — 1 

the reason being apparent on inspection. 

In numerical problems under Eq. (9) there will be known 

P2V2, n, and either pi or Vi. The unknown must be computed 

from the relations from'Eq. (5) : 

1 

Vi = P2 - or Vi = V 2 ,i 



v\l \Pi 

Example 3a. A compressed-air motor takes air at a gage 
pressure = 100 lbs. and works with a cut-off at \ stroke. 
What work (ft. -lbs.) will be gotten per cu. ft. of compressed 
air, assuming free air pressure = 14.5 lbs. and n = 1.41 ? 

Solution. Applying Eq. (9) and noting that all pressures 

are to be multiplied by 144 and that the pressure at end of 

/iy.41 
stroke = pi = 114.5 f^j = 16.3 and that Vi = £v 2 , we get 

TF = 144^ 114 - 5X1 ~ 16 - 3X4 +114.5X1-14.5X4^= 25,444. 

Art. 4. Effect of Clearance : In Compression. 

It is not practicable to discharge all of the air that is 
trapped in the cylinder; there are some pockets about the 
valves that the piston cannot enter, and the piston must not 
be allowed to strike the head of the cylinder. This clearance 
can usually be determined by measuring the water that can 
be let into the cylinder in front of the piston when at the end 
of its stroke; but the construction of each compressor must 



FORMULAS FOR WORK 9 

be studied before this can be undertaken intelligently, and 
it is not done with equal ease in all machines. 

To formulate the effect of this clearance in the operation 
of the machine, 

Let v = volume of piston displacement ( = area of piston 
X length of stroke) , 

Let cv = clearance, c being a percentage. 

Then v + c v is the volume compressed each stroke. But 
the clearance volume cv will expand to a volume rev as the 
piston recedes, so that the fresh air taken in at each stroke 
will be v + cv — rev, and the volumetric efficiency will be 

™ v + cv — rev ., , /-, n /irk x 

E v = — ! = l + c(l— r). (10) 

v 

When E v = c = and no air will be discharged. 

r — 1 

Theoretically (as the word is usually used) clearance does 
not cause a loss of work, but practically it does, insomuch as 
it requires a larger machine, with its greater friction, to do 
a given amount of effective work. 

Example 4a. A compressor cylinder is 12" diam. X 16" 
stroke. The clearance is found to hold 1J pints of water 

X 231 =36 cubic inches, therefore c = 



8 ' 113 X 16 

= 0.02. 
Then by Eq. (10) when r = 7 

E = 1 + 0.02 (1-7)= 88%. 

Such a condition is not abnormal in small compressors, and 
the volumetric efficiency is further reduced by the heating 
of air during admission as considered in Art. 6. 

Art. 5. Effect of Clearance and Compression in Expansion 
Engines. 

Fig. 4 is an ideal indicator diagram illustrating the effect 
of clearance and compression in an expansion engine. 

In this diagram the area E shows the effective work, D 
the effect of clearance, B the effect of back pressure of the 
atmosphere and C the effect of compression on the return 
stroke. 



10 



COMPRESSED AIR 



The study of effect of clearance in an expansion engine 
differs from the study of that in compression, due to the 
fact that the volume in the clearance space is exhausted 
into the atmosphere at the end of each stroke. 




j*— bl— +{ 



Fig. 4. 



If the engine takes full pressure throughout the stroke the 
air (or steam) in the clearance is entirely wasted ; but when 
the air is allowed to expand as illustrated in the diagram some 
useful work is gotten out of the air in the clearance during 
the expansion. 

The loss due to clearance in such engine is modified by the 
amount of compression allowed in the back stroke. If the 
compression p c = p 2 , the loss of work due to clearance will be 
nothing, but the effective work of the engine will be consid- 
erably reduced, as will be apparent by a study of a diagram 
modified to conform to the assumption. 

While the formula for work that includes the effect of 
clearance and compression will not be often used in practice 
its derivation is instructive and gives a clear insight into 
these effects. 

The symbols are placed on the diagram and will not need 
further definition. 

The effective work E will be gotten by subtracting from 
the whole area the separate areas B, C and D. From Art. 2, 
after making the proper substitutions for the volumes, there 
results 

Total area = 1 [ *» < c +^-^ (1 + c) + p 2 (c + k)\ 



FORMULAS FOR WORK 11 

Area B = lp a , 

Area D = lp 2 c, 

Area C = ll VcC -pAb^ + c) _ n 

Subtracting the last three from the first and reducing their 
results : 

—JT~ = z[c(p2 + Pa-Pc-Pl)+n(P2k + pJ>-Pa)-(Pl-Pa)] 

At n—1 

= Mean effective pressure. 

The actual volume ratio before and after expansion is 

V2 _ cvi -f- kvi _ c -\-k 

Vi CVi -\-V\ c + 1 

This is the ratio with which to enter Table I to get r and t 
and from r the unknown pressure pi. Similarly for the 

compression curve the ratio of volumes is -, and pc can be 

o 

found as indicated above. 

Art. 6. Effect of Heating Air as it Enters Cylinders. 

When a compressor is in operation all the metal exposed 
to the compressed air becomes hot even though the water 
jacketing is of the best. The entering air comes in contact 
with the admission valves, cylinder head and walls and the 
piston head and piston rod, and is thereby heated to a very 
considerable degree. In being so heated the volume is in- 
creased in direct proportion to the absolute temperature 
(see Eq. (5)), since the pressure may be assumed constant 
and equal that of the atmosphere. Hence a volume of 
cool free air less than the cylinder volume will fill it when 
heated. This condition is expressed by the ratio 

^ = l f or v a = vjf, 

where v c and t c represent the cylinder volume and tempera- 
ture. The volumetric efficiency as effected by the heating is 

77T V a t a 



12 COMPRESSED AIR 

Example 6a. Suppose in Example 4a the outside free air 
temperature is 60° F. and in entering the temperature rises 
to 160° F., then 

t a 460 + 60 QA . 

- — -77^ — — ™ = 84 per cent. 
t c 460 + 160 

Then the final volumetric efficiency would be 88 X 84 = 
74% nearly. 

The volumetric efficiency of a compressor may be further 
reduced by leaky valves and piston. 

In Arts. 4 and 6 it is made evident that the volumetric 
efficiency of an air compressor is a matter that cannot be 
neglected in any case where an installation is to be intelli- 
gently proportioned. It should be noted that the volu- 
metric efficiency varies with the various makes and sizes 
of compressors and that the catalog volume rating is always 
based on the piston displacement. 

These facts lead to the conclusion that much of the uncer- 
tainty of computations in compressed-air problems and the 
conflicting data recorded is due to the failure to determine 
the actual amount of air involved either in terms of net 
volume and temperature or in pounds. 

Methods of determining volumetric efficiency of air com- 
pressors are given in Chapter III. 

The loss of work due to the air heating as it enters the 
compressor cylinder is in direct proportion to the loss of 
volumetric efficiency due to this cause. In Example 6a 
only 84% of the work done on the air is effective. 

By the same law any cooling of the air before entering the 
compressor effects a saving of power. See Art. 9. 

Art. 7. Change of Temperature in Compression or Ex- 
pansion. 

Eq. (4) may be written 

PlVi = cti; p 2 v 2 = ct 2 

and Eq. (5) may be factored thus, 

P1V1V1 1 - 1 = p 2 v 2 v 2 n ~ l . 

Substituting we get 

ctid 71 - 1 = ct 2 v 2 n ~K 



FORMULAS FOR WORK 13 



Whence 


*-'&-' 


(ii) 


and 


n-l 


(Ha) 


since from Eq. (5) 


1 





^2 XPV 

It is possible to compute n from Eq. (11) by controlling the 
Vi and V2 and measured t\ and 2 2 - 

Table I, columns 5 and 6, is made up from Eq. (11a) and 
columns 3 and 4 from Eq. (5) as just written. 

Example 7. What would be the temperature of air at the 
end of stroke when r = 7 and initial temperature = 70° F.? 

Solution. Referring to Table I in line with r = 7 note that 

1.4758 when n = 1.25 
tj ^ /. k = (460 + 70) X 1.4758 - 460 = 322° F. 

h 1.7585 when n = 1.41 

/. t 2 = (460 + 70) X 1.7585 - 460 = 472° F. 

From the same table the volume of one cubic foot of free 
air when compressed and still hot would be respectively 0.21 
and 0.25, while when the compressed air is cooled back to 
70° its volume would be 0.143. 

Art. 8. Density at Given Temperature and Pressure. 

By Eq. (4) pv = 53.17 t for one pound, and the weight of 
one cubic foot 

= w = l=—P — (12) 

v 53.17 1 K } 

Note that p must be the absolute pressure in pounds per 
square foot, and t the absolute temperature. When gage 
pressures are used and ordinary Fahrenheit temperature 
the formula becomes 

53.17 V460 + F/ 

- 2 - 7o i^f- F } (i2a) 

Table III is made up from Eq. (12). 



14 COMPRESSED AIR 

Art. 9. Cooling Water Required. 

In isothermal changes, since pv is constant, evidently 
there is no change in the mechanical energy in the body of 
air as measured by the absolute pressure and using the term 
"mechanical energy " to distinguish from heat energy. Hence 
evidently all the work delivered to the air from outside must 
be abstracted from the air in some other form, and we find 
it in the heat absorbed by the cooling water. Therefore, 

2^1^: = (B.T.U's) 

780 v } 

of work done on compressed air = 35.5 log r (B.T.U's) per 
pound of air compressed from temperature of 60° F. If the 
water is to have a rise of temperature T° (T being small, else 
the assumption of isothermal changes will not hold), then 

- &— = Pounds of water required in same time. 

780 T H 

Example 8a. How many cubic feet of water per minute will 
be required to cool 1000 cubic feet of free air per minute, 
air compressed from p a = 14.2 to p g = 90° gage, initial tem- 
perature of air = 50° F. and rise in temperature of cooling 
water = 25° ? 

Solution: 

144 X (90+14.2) X 1000 Xlog/ 90 ^ 4,2 ) 

— — — -- — „^ „ : = 24 cu. ft. per min. 

780 X 25 X 62.5 F 

It is practically possible to attain nearly isothermal con- 
ditions by spraying cool water into the cylinder during 
compression. In such a case this article would enable the 
designer to compute the quantity of water necessary and 
therefrom the sizes of pipes, pumps, valves, etc. 

Art. 10. Reheating and Cooling. 

In any two cases of change of state of a given weight of 
air, assuming the ratio of change in pressure to be the same, 
the work done (in compression or expansion) will be directly 
proportional to the volume, as will be evident by examina- 
tion of the formulas for work. Also at any given pressure 
the volumes will be directly proportional to the absolute tem- 
peratures. Hence the work done either in compression or 



FORMULAS FOR WORK 15 

expansion (ratio of change in pressures being the same in each 
case) will be directly proportional to the absolute initial tem- 
peratures. 

Thus if the temperature of the air in the intake end of one 
compressor is 160° F. and, in another 50° F., the work done 
on equal weights of air in the two cases will be in the pro- 
portion of 460 + 150 to 460 + 50, or 1.2 to 1; that is, the 
work in the first case is 20% more than that in the second 
case. This is equally true, of course, for expansion. 

The facts above stated reveal a possible and quite practi- 
cable means of great economy of power in compressing air 
and in using compressed air. 

The opportunities for economy by cooling for compression 
are not as good as in heating before the application in a 
motor, but even in compression marked economy can be 
gotten at almost no cost by admitting air to the compressor 
from the coolest convenient source, and by the most thorough 
water-jacketing with the coolest water that can be conven- 
iently obtained. 

In all properly designed compressor installations the air is 
supplied to the machine through a pipe from outside the 
building to avoid the warm air of the engine room. In 
winter the difference in temperature may exceed 100°, and 
this simple device would reduce the work of compression by 
about 20%. For the effect of intercoolers and interheaters 
see Art. 12 on compounding. 

By reheating before admitting air to a compressed-air 
engine of any kind the possibilities of effecting economy 
of power are greater than in cooling for compression, the 
reason being that heating devices are simpler and less costly 
than any means of cooling other than those cited above. 

The compressed air passing to an engine can be heated to 
any desired temperature; the only limit is that temperature 
that will destroy the lubrication. Suppose the normal 
temperature of the air in the pipe system is 60° F. and that 
this is heated to 300° F. before entering the air engine, then 
the power is increased 46%. Reheating has the further 
advantage that it makes possible a greater ratio of expansion 
without the temperature reaching freezing point. 



16 



COMPRESSED AIR 



The devices for reheating are usually a coil or cluster of 
pipes through which the air passes while the pipe is exposed 
to the heat of combustion from outside. Ordinary steam 
boilers may be used, the air taking the place of the steam and 
water. 

Unwin suggests reheating the air by burning the fuel in 
the compressed air as suggested in the cut. 



Gas, Liquid o r -^- j 
Powdered Fuel ** I — m 



Cold Air 



"l 



Combustion 



i 



Hot Air 



Fig. 4 A. 

Even when the details are worked out such a device would 
be simple and inexpensive. The theoretic advantages of 
such a device are that all the heat would go into the air, 
the gases of combustion (if solid or liquid fuel be used) would 
increase the volume, and the combustion occurring in com- 
pressed air would be very complete. 

The author has no knowledge of any such devices having 
been used in practice. 

The power efficiency of the fuel used in reheaters is very 
much greater than that of the fuel used in steam boilers. 
Unwin states that it is five or six times as much. The chief 
reason is that none of the heat is absorbed in evaporation 
as in a steam boiler. 

In many of the applications of compressed air reheating is 
impracticable, and efficiency is secondary to convenience — 
but in large fixed installations, such as mine pumps, reheat- 
ing should be applied. 

Art. 11. Compounding. 

In steam-engine designs compounding is resorted to to 
economize power by saving steam, while in air compressors 
and compressed-air engines compounding is resorted to for 
the twofold purpose of economizing power and controlling 
temperature, both objects being accomplished by reducing 
the extreme change of temperature. The economic prin- 



FORMULAS FOR WORK 



17 



ciples involved in compound steam engines and in com- 
pound air engines are quite different, the reasons underlying 
the latter being much more definite. 

The air is first compressed to a moderate ratio in the 
low-pressure cylinder, whence it is discharged into the " inter- 
cooler," where most of the heat developed in the first stage 
is absorbed and thereby the volume materially reduced, so 
that in the second stage there will be less volume to com- 
press and a less injurious temperature. 

The changes occurring and the manner in which economy 
is effected in compression may be most easily understood 
by reference to Fig. 5, which represents ideal indicator 
diagrams from the two cylinders, superimposed one over 
the other, the scale being the same in each, the dividing 
line being kb. 




Fig. 5. 

In this diagram, Fig. 5, 

dbk is the compression line in the first-stage or low-pressure 
cylinder, 

cds is the compression line in the second-stage or high-pres- 
sure cylinder, 

be is the reduction of volume in the intercooler, 



18 



COMPRESSED AIR 



aby would be the pressure line if no intercooling occurred, 

The area cdfb is the work saved by the intercooler, 

ace would be the compression line for isothermal compres- 
sion, 

aug would be the compression line for adiabatic compres- 
sion. 

The diagram Fig. 5 is correctly proportioned for r = 6. 

Fig. 6 is a diagram drawn in a manner similar to that used 
in Fig. 5 and is to illustrate the changes and economy effected 
by compounding with heating when compressed air is applied 
in an engine. It is assumed that the air is " preheated, " 
that is, heated once before entering the high-pressure cylinder 
and again heated between the two cylinders. 




W " fu 



Fig. 6. 



In this diagram, Fig. 6, 
se is the volume of compressed air at normal temperature, 
sf is the volume of compressed air after preheating, 
fc is the expansion line in the high-pressure cylinder, 
cb is the increase of volume in the interheater, 
by is the expansion line in low-pressure cylinder, 

ezq would be the adiabatic expansion line without any 
heating, 

efcz is work gained by preheating, 

cbyx is work gained by interheating. 



FORMULAS FOR WORK 19 

In no case is it economical to expand down to atmospheric 
pressure. Hence the diagram is shown cut off with pressure 
still above that of free air. 

The diagram Fig. 6 is proportioned for preheating and re- 
heating 300° F. 

Art. 12. Proportions for Compounding. 

It is desirable that equal work be done in each stage of 
compounding. If this condition be imposed, Eq. (8) indi- 
cates that the r must be the same in each stage, for on the 
assumption of complete intercooling the product pv will 
be the same at the beginning of each stage. 

If then ri be the ratio of compression in the first stage, 
the pressure at end of first stage will be rip a = p h and the 
pressure at end of second stage = ripi = r 2 p a — p 2 , and 
similarly at end of third stage the pressure will be p 3 = r x 3 p a , 
or 

In two-stage work n = l^ 1 ) = r 2 . 

xpj 



In three-stage work r± = I ^ ) = r 3 *. 

XPJ 



Let V\ = free air intake per stroke in low-pressure cylinder 
or first stage, 
v 2 = piston displacement in second stage, 
v z = piston displacement in third stage, 
ri = ratio of compression in each cylinder. 
Then, assuming complete intercooling, 

v 2 = - and v 3 = - = —. 

or — = — and — = — -• 

The length of stroke will be the same in each cylinder; 
therefore the volumes are in the ratio of the squares of 
diameters, or 



£-1 and «-X 

di 2 ri di 2 ri 2 

Hence d 2 = -~ and d z = — • (12b) 



20 COMPRESSED AIR 

If the intention to make the work equal in the different 
cylinders be strictly carried out it will be necessary to make 
the first-stage cylinder enough larger to counteract the 
effect of volumetric efficiency. Thus if volumetric efficiency 
be 75%, the volume (or area) of the intake cylinder should 
be one-third larger. Note that the volumetric efficiency is 
chargeable entirely to the intake or low-pressure cylinder. 
Once the air is caught in that cylinder it must go on. 

Example 12. Proportion the cylinders of a compound two- 
stage compressor to deliver 300 cu. ft. of free air per minute 
at a gage pressure = 150. Free air pressure = 14.0, 
R.P.M. = 100, stroke 18", piston rod If" diameter, volu- 
metric efficiency = 75%. 

Solution. From the above data the net intake must be 
3 cu. ft. per revolution. Add to this the volume of one piston 
rod stroke ( = .025 cu. ft.) and divide by 2. This gives 
the volume of one piston stroke 1.512. The volume of one 

foot of the cylinder will be — X 1.512 = 1 . 008 cu. ft. From 

Table X the nearest cylinder is 14" diam., the total ratio of 

150 -4- 14 
compression = y — = 11.71, and the ratio in each stage 

is (11.71)*= 3.7 = r h and by (12b) 

d 2 = — ~ = = 7.3", say 7f", for the high-pressure cylinder. 

(ri) 2 1-92 

Now we must increase the low-pressure cylinder by one- 
third to allow for volumetric efficiency. The volume per 
foot will then be 1.344, which will require a cylinder about 
15f " diameter. Note that the diameter of the high-pressure 
cylinder will not be affected by the volumetric efficiency. 

Art. 13. Work in Compound Compression. 

Assuming that the work is the same in each stage, Eq. (8) 
can be adapted to the case of multistage compression thus : — 

In two-stage work 

-A(r^-l)x2 (13) 



n — 



n 



n — 1 



) a Va\r2 2n -V 



VaVaW n -1/X2. (13a) 



FORMULAS FOR WORK 21 

In three-stage work 

■WaW^-ljxi (13b) 



n — 1 



n 



WaVt 3U ~ l) 



,PaVaW n -1/X3. (13C) 

n — 1 
Note that r 2 = ^ and r 3 = ^ and also that p a v a = p&i = 

Pa Pa 

P2V2, etc., assuming complete intercooling. 

Laborious precision in computing the work done on or by 
compressed air is useless, for there are many uncertain and 
changing factors: n is always uncertain and changes with 
the amount and temperature of the jacket water, the volu- 
metric efficiency, or actual amount of air compressed, is 
usually unknown, the value of p a varies with the altitude, 
and r is dependent on p a . 

Art. 14. Work under Variable Intake Pressure. 

There are some cases where air compressors operate on air 
in which the intake pressure varies and the delivery pressure 
is constant. This is true in case of exhaust pumps taking air 
out of some closed vessels and delivering it into the atmos- 
phere. It is also the condition in the " return-air" pumping 
system in which one charge of air is alternately forced into 
a tank to drive the water out and then exhausted from the 
tank to admit water. For full mathematical discussion of 
this pump see Trans. Am. So. C. E., Vol. 54, p. 19. The 
following formulas and others more complex were first 
worked out to apply to that pumping system. 

In such cases it is necessary to determine the maximum 
rate of work in order to design the motive power. 

First assume the operation as being isothermal. Then 
in Eq. (1), viz. 

W = p x V loge ^ 

Px 

p x is variable, while v and p\ are constant. In this formula 
W becomes zero when p x is zero and again when p x — pi, 
since log 1 is zero. To find when the work is maximum, 
differentiate and equate to zero; thus differential of 



22 COMPRESSED AIR 

v (p* log Pi-Px log Vx) =v\ log pidp x - ( p x -^ + \og p x dp x A 
Equate this to zero and get log pi = 1 + log p x , 
or log. 21 = 1, therefore ^ = e = 2.72. 

Px Px 

That is, when r = 2.72 the work is a maximum. 

When the temperature exponent n is to be considered the 
study must be made in Eq. (8), viz. 

n-l 



Differentiating this with respect to p x and equating to zero, 



TT7- w 

Tr = — p x v 

n — 1 



-i 

71 



the condition for maximum work becom 3s[ — ) = n. Insert 

\Pxl 

this in (8) and the reduced formula becomes 

W = n pxv. = — J— 



From the above expressions for maximum the following 
results : 

When n = 1.41 the maximum occurs when r = 3.26. 
When n = 1.25 the maximum occurs when r = 3.05. 
When n = 1. the maximum occurs when r = 2.72. 
In practice r = 3 will be a safe and convenient rule. 

Exercise 14a. Air is being exhausted out of a tank by an ex- 
haust pump with capacity = 1 cu. ft. per stroke. At the be- 
ginning the pressure in the tank is that of the atmosphere = 
14.7 lbs. per sq. in. Assume the pressure to drop by intervals 
of one pound and plot the curve of work with p x as the 
horizontal ordinate and W as the vertical, using the formula 

W = p x vlog^- 

Px 

Exercise 146. As in 14a plot the curve by Eq. (8) with 
n = 1.25. 

Art. 15. Exhaust Pumps. 

In designing exhaust pumps the following problems may 
arise. 



FORMULAS FOR WORK 23 

Given a closed tank and pipe system of volume V under 
pressure p and an exhaust pump of stroke volume v, how 
many strokes will be necessary to bring the pressure down 
to p m 1 

The analytic solution is as follows, assuming isothermal 
conditions in the volume V. 

The initial product of pressure by volume is p V. After 
the first stroke of the exhaust pump this air has expanded 
into the cylinder of the pump and pressure has dropped to 
Pi under the law that pressure by volume is constant. 

Hence (V + v) p\ = PoV, or »i = -^ — at end of first stroke, 

V + v 

(V + v)p 2 = PiV, or p 2 = ^— = Vo(t^—- ) 

V + v \ V + vj 

at end of second stroke, 

(V + v)ps = p 2 V, or p 3 = p 2 ——-==pJ——Y 

V + v \ V + v/ 

at end of third stroke, etc. 

log &* 
Finally Vm = pJ-Y—Y and m= ffj> (14) 

m is the required number of strokes. 

Example 15a. A closed tank containing 100 cu. ft. of air 
at atmospheric pressure (= 14.5 lbs. per sq. in.) is to be 
exhausted down to 5 lbs. by a pump making 1 cu. ft. per 
stroke. How many strokes required ? 

Solution. 2Zzz* _ . an( j _ _. ., 

p 14.5 V + v 101 

log 5 = 0.69897 log 100 = 2.00000 

log 14.5 = 1.16136 log 101 = 2.00432 

1.53761 1.99568 

These two logarithms may be written thus: 

- 1 + 0.53761 = - .46239 -, .46239 irV7 - 

and = 107 = m. 

- 1 + 0.99568 = - .00432 .00432 

If the volumetric efficiency of the machine be E, then the 
number of strokes would be 107 -f- E. 



24 COMPRESSED AIR 

The results found under Arts. 14 and 15 serve well to 
illustrate the curious mathematical gymnastics that com- 
pressed air is subject to, and should encourage the investi- 
gator who likes such work, and should put the designer on 
guard. 

Art. 16. Efficiency when Air is Used without Expansion. 

In many applications of compressed air convenience and 
safety are the prime requisites, so that power efficiency 
receives little attention at the place of application. This 
is so with such apparatus as rock drills, pneumatic hammers, 
air hoists and the like. The economy of such devices is so 
great in replacing human labor that the cost in power is 
little thought of. Further, the necessity of simplicity and 
portability in such apparatus would bar the complications 
needed to use the air expansively. There are other cases, 
however, notably in pumping engines and devices of various 
kinds, where the plant is fixed, the consumption of air con- 
siderable and the work continuous, where neglect to work 
the air expansively may not be justified. 

In any case the designer or purchaser of a compressed-air 
plant should know what is the sacrifice for simplicity or low 
first cost when the proposition is to use the air at full pres- 
sure throughout the stroke and then exhaust the cylinder 
full of compressed air. 

Let p be the absolute pressure on the driving side of the 
piston and p a be that of the atmosphere on the side next 
the exhaust. Then the effective pressure is p — p a and the 
effective work is (p — p a ) v, while the least possible work 
required to produce this air is pv log e r. 

Hence the efficiency is E = ^ ^ a ' • 

pv log e t 

Dividing numerator and denominator by p a v this reduces to 

E=^=±. (15) 

r log e r 

This is the absolute limit to the efficiency when air is used 
without expansion and without reheating. It cannot be 
reached in practice. 

Table VI represents this formula. Note that the effi- 



FORMULAS FOR WORK 25 

ciency decreases as r increases. Hence it may be justi- 
fiable to use low-pressure air without expansion when it 
would not be if the air must be used at high pressure. 

Clearance in a machine of this kind is just that much 
compressed air wasted. If clearance be considered, Eq. (15) 
becomes 

E = n /~ 1 1 (15a), 

(l+c)rlog e r 

where c is the percentage of clearance. In some machines, 
if this loss were a visible leak, it would not be tolerated. 

Art. 17. Variation of Atmospheric Pressure with Altitude. 

In most of the formulas relating to compressed-air opera- 
tions the pressure p a , or weight w a , of free air is a factor. 
This factor varies slightly at any fixed place, as indicated 
by barometer readings, and it varies materially with varying 
elevations. 

To be precise in computations of work or of weights or 
volumes of air moved, the factors p a and w a should be deter- 
mined for each experiment or test, but such precision is 
seldom warranted further than to get the value of p a for 
the particular locality for ordinary atmospheric conditions. 
This however should always be done. It is a simple matter 
and does not increase the labor of computation. In many 
plants in the elevated region p a may be less than 14.0 lbs. 
per sq. in., and to assume it 14.7 would involve an error of 
more than 5%. 

Direct reading of a barometer is the easiest and usual way 
of getting atmospheric pressure, but barometers of the 
aneroid class should be used with caution. Some are 
found quite reliable, but others are not. In any case they 
should be checked by comparison with a mercurial barom- 
eter as frequently as possible. 

If m be the barometer reading in inches of mercury and 
F be the temperature (Fahrenheit), the pressure in pounds 
per sq. in. is 

P* = ^|pm[l-.0001(F-32)] 

= ,4931 m [1 - .0001 (F - 32)]. (16) 



26 COMPRESSED AIR 

The information in Table II will usually obviate the need of 
using Eq. (16). 

In case the elevation is known and no barometer available 
the problem can be solved as follows: 

Let p s = pressure of air at sea level, 
w s = weight of air at sea level, 
p x , w x be like quantities for any other elevation. 

Then in any vertical prism of unit area and height dh we 
have 

Vx + dp x = p x + w x dh, 

or dp x = w x dh. 

But — - = ^2; therefore dp x = — p x dh, 

™s Ps Ps 

or dh = — -^, and therefrom h = 2s. x log 2i f 

W s p x W s p a 

where pa is the pressure at elevation h above sea level. Sub- 
stitute for w s its equivalent 

w '~swt and we get ah = log p. • 

Whence log e p a = log e p s — 



53.17 * 



Making ps = 14.745 and adopting to common logarithm 
and Fahrenheit temperatures, 

lo gl0 p„ = 1.16866- mA £ +m - W 

Table V is made up by formula 17. 



CHAPTER II 

Measurement of Air 

Art. 18. General Discussion. 

Progress in the science of compressed-air production and 
application has evidently been hindered by a lack of accu- 
rate data as to the amount of compressed air produced and 
used. 

The custom is almost universal of basing computations 
on, and of recording results as based on, catalog rating of 
compressor volumes — that is, on piston displacement. 

The evil would not be so great if all compressors had 
about the same volumetric efficiency, but it is a fact that the 
volumetric efficiency varies from 60 per cent to 90 per cent, 
depending on the make, size, condition and speed of the 
machine; no wonder, then, that calculations often go wrong 
and results seem to be inconsistent. 

There are problems in compressed-air transmission and use 
for the solution of which accurate knowledge of the volume 
or weight of air passing is absolutely necessary. Chief among 
these are the determination of friction factors in air pipes 
and the efficiency of pumps, including air lifts. 

Purchasers may be imposed upon, and no doubt often are, 
in the purchase of compressors with abnormally low volu- 
metric efficiencies. Contracts for important air-compressor 
installation should set a minimum limit for the volumetric 
efficiency, and the ordinary mechanical engineer should have 
knowledge and means sufficient to test the plant when 
installed. 

There is little difficulty in the measurement of air. The 
calculations are a little more technical, but the apparatus 
is as simple and the work much less disagreeable than in 
measurements of water. 

In many text-books theoretic formulas are presented for 

27 



28 



COMPRESSED AIR 



the flow of air at high pressures through orifices into the 
atmosphere. Such formulas are complicated by the neces- 
sity of considering change of volume and temperature, and 
even where the proper empirical coefficients are found the 
formulas are unwieldy. 

Art. 19. Apparatus for Measuring Air by Orifice. 

Present indications are that the standard method of 
determining flow of air will require the pressure to be reduced 
to less than one foot head of water in order that change 
of volume and temperature may be neglected and the for- 
mula simplified thereby. 

Experiments under such circumstances show coefficients 
even more constant than those for standard orifices for 
measuring water. The coefficients given in Art. 20 were 
determined at McGill University by methods and apparatus 
described first in Trans. Am. So. Mech. E., Vol. 27, Dec, 1905, 
and later in Compressed Air, Sept., 1906, p. 4187. 

Having the coefficients once determined, the necessary 
apparatus is simple and inexpensive. The essentials are 
shown in Fig. 7. 




Fig. 7. 



A = Compressed-air pipe, 

B = Closed box or cylinder, 

T = Throttle, 

b = Baffle boards or screen, 

H = Thermometer, 

C = Cork, 

= Orifice in thin metal plate (Standard), 

U = Bent glass tube containing colored water, 

G = Scale of inches. 



MEASUREMENT OF AIR 29 

The box B may be made of any convenient light material. 
The pressure is only a few ounces and the tendency to leak 
slight. The purpose of the throttle T is to control the 
pressure against which the compressor works. The appro- 
priate orifice can be determined by a preliminary com- 
putation, assuming h at say 5". See Eq. (14). 

In testing a compressor it should be run until every part 
is at its normal running temperature. By means of the 
throttle T the compressor can be worked under various 
pressures and speed and thereby its individual curves of 
volumetric efficiency obtained. 

Art. 20. Formula for Standard Orifice under Low 
Pressure. 

Let p a = air pressure in lbs. per sq. in. inside the box, 
Q = weight of air passing per second, 
w = weight of a cubic foot of air in pounds, 
d = diameter of orifice in inches, 
i — pressure as read on water gage in inches, 
t = absolute temperature Fahrenheit's scale, 
c = the experimental coefficient. 

Where changes of density and temperature can be neg- 
lected the theoretic velocity through the orifice is V = V2 gh, 
where h is the head of air of uniform density that would 
produce the pressure i. 

Hence h = — X — ; therefore v = J 2 g — X — • 
12 w a V 12 w a 

But Q = w a Xav where a = area of orifice in sq. ft. = 

4X144 

Inserting these values and putting w a under the radical there 
results 

^ " 4 X 144 V g 12 w a 



But W a = 



Pa 



53.17 * 



Q = .0136d 2 yA Va ' = .1632 d 2 \Ji p a (18) 



r > 

where p a is in lbs. per sq. in. 



30 COMPRESSED AIR 

The pressure due to i ( = .036 i) should be included in p a . 
If the work is at sea level and pressure i be neglected, 
p a = 14.7 X 144 and the formula becomes 

Q = .6299 d 2 v/-, (18a) 

which is the formula published by McGill University. 

This is the theoretic formula. To it must be applied the 
experimental coefficient c as given in Table VIII. Note 
that c varies but little from 0.60, and the same c can be 
used in Eq. (18) and (18a). 

Example 20a. In a run with the apparatus shown in 
Pig. 7 the following was recorded: d = 2.32"; i = 4.6"; 
T = 186° F. inside drum, T= 86° F. in free air. Elevation 
1200'. Find the weight and volume of free air passing. 

Solution. From Table II, interpolating for 86° in the line 
-with 1200 elevation we get w a =.0700 and p a for free air 
= 14.1. Add the pressure due to i ( = .036 X 4.6) and we 
get the corrected p a = 14.26. In Table VIII the coefficient 
for d = 2.32 and i = 4.6 is 0.599. These numbers inserted 
in (18) give 

Q = .599 X .1632 X (2.32) 2 v/^ X 14.26 

▼ 54b 

= . 1832 pound per second 
and the free air volume 

= '- X 60 = 157 cu. ft. per minute. 

.0700 F 

By Eq. (18a) without corrections Q = .1857. 

Art. 21. Air Measurement in Tanks. 

The amount of air put into or taken out of a closed tank 
or system of tanks and pipes, of known volume, can be 
accurately determined by Eq. (3), viz., 

VaVa _ \a „„ .. _ Px^a Vx m 
Px^x tx Va tx 

By this means the volume of air delivered into a closed sys- 
tem by a compressor can be very accurately determined. 

The process would be as follows : Determine the volumes 
of all tanks, pipes, etc., to be included in the closed system, 



MEASUREMENT OF AIR 31 

open all to free air and observe the free-air temperature; 
then switch the delivery from the compressor into the 
closed system; count the strokes of the compressor until the 
pressure is as high as desired; then shut off the closed tank 
and note pressure and temperatures of each separate part 
of the volume. Then the formula above will give the vol- 
ume of free air which compressed and heated would occupy 
the tanks. From this subtract the volume of free air origi- 
nally in the tanks; the remainder will be what the compressor 
has delivered into the system. Note that the compressor 
should be running hot and at normal speed and pressure 
when the test is made for its volumetric efficiency. 

Usually the temperature changes will be considerable, but 
if the system is tight, time can.be given for the temperature 
to come back to that of the atmosphere, thus saving the 
necessity of any temperature observations. 

Where a convenient closed-tank system is available this 
method is recommended. 

This method — that is, Eq. (3) as stated above — was used 
to determine the quantity of air passing the orifices in the 
experiments by which the coefficients were determined as 
given in Art. 20, Table VII. 

Example 21a. A- tank system consists of one receiver 3' 
diam. X 12', one air pipe 6" X 40', one 4" X 4000'' and a 
second receiver at end of pipe 2' diam. X 8'. A compres- 
sor 12" X 18" with \\" piston rod puts the air from 1250 
revolutions into the system, after which the pressure is 
80 gage and temperature in first receiver 200°, while in 
other parts of the tank system it is 60°. Temperature of 
outside air being 50°, p a = 14.5 per sq, in. Find volu- 
metric efficiency of the compressor. 

Solution. Volumes (from Table X) : 

1st receiver 84.84 cu. ft. 

6" pipe 7.84 ) 

4" pipe 349.20 } 382.16 

2nd receiver 25.12 J 

Total 467.00 in tank system. 



32 COMPRESSED AIR 

Piston displacement in one revolution = 2.338 cu. ft. 
(piston rod deducted). 

By formula v a = (2-_ ) X - note that the quantity in paren- 
V Pa I t x 

thesis is constant and therefore a slide rule can be conven- 
iently used, otherwise work by logarithms. 

... • (8 0+14.5) (460+50) w 84.84 ,.- 
«. m first receiver = L -21 1 x _____ = 417.2 

va in 6" pipe, 4" pipe and second receiver with total 

volume 382.16 and t = 60° = 2447.1 

Total 2864.3 

Original volume of free air 467 

Volume of free air added 2397.3 

2397.3 4- 2.338 = 1028. 

Therefore the volumetric efficiency is 

E = 1028 -T- 1250 = 82%. 



CHAPTER III 

Friction in Air Pipes 

Art. 22. In the literature on compressed air many for- 
mulas can be found that are intended to give the friction in 
air pipes in some form. Some of these formulas are, by 
evidence on their face, unreliable, as for instance when no 
density factor appears; the origin of others cannot be traced 
and others are in inconvenient form. Tables claiming to 
give friction loss in air pipes are conflicting, and reliable 
experimental data relating to the subject are quite limited. 

In this article and the next are presented the derivation 
of rational formulas for friction in air pipes with full exposi- 
tion of the assumptions on which they are based. The coeffi- 
cients were gotten from the data collected in Appendix B. 

Art. 23. The Formula for Practice. 

The first investigation will be based on the assumption that 
volume, density and temperature remain constant through- 
out the pipe. 

Evidently these assumptions are never correct; for any 
decrease in pressure is accompanied by a corresponding 
increase in volume even if temperature is constant. (The 
assumption of constant temperature is always permissible.) 
However, it is believed that the error involved in these 
assumptions will be less than other unavoidable inaccuracies 
involved in such computations. 

Let / = lost pressure in pounds per sq. in., 
I = length of pipe in feet, 
d = diameter of pipe in inches, 
s = velocity of air in pipe in feet per second, 
r = ratio of compression in atmospheres, 
c = an empirical coefficient including all constants. 

33 



34 COMPRESSED AIR 

Experiments have proved that fluid friction varies very 
nearly with the square of the velocity and directly with 
the density. Hence if k be the force in pounds necessary 
to force atmospheric air (r = 1) over one square foot of sur- 
face at a velocity of one foot per second, then at any other 
velocity and ratio of compression the force will be 

Fi = Mr, 

and the force necessary to force the air over the whole 
interior of a pipe will be 

F=—lXkrs 2 , 

and the work done per second, being force multiplied by 
distance, is 

Work = — X krs B . 
12 

Now if the pressure at entrance to the pipe is / pounds per 
sq. in. greater than at the other end, the work per second 
due to this difference (neglecting work of expansion in air) is 

Work = f^—- s. 

J 4 

Equating these two expressions for work there results 

/. ird 2 ird 7 7 o 

f — s = — Ikrs 6 , 
J 4 12 

or / = — k - rs 2 . (19) 

Now the volume of compressed air, v, passing through the 
pipe is, in cubic feet, 



v = 



■d 2 



4X 144 
and the volume of free air v a is rv. 

Therefore v a = — X rs 

4 X 144 

and , 2= (4X144)V 

7rW 



FRICTION IN AIR PIPES 



35 



Insert this value of s 2 in Eq. (19) and reduce and there results 



or 



J ' 12 V 7T / d 5 r 

f =c To' 
a r 



(20) 



where c is the experimental coefficient and includes all 
constants. 

clv a 2V 



From Eq. (20), 



d = 



fr 



(21) 



From the data collected in Appendix B the following 
results were computed. In this r and s are mean results 
and c is the average of all the runs made on each pipe. 



d 


c 


r 


s 


t 


h 


.092 


2. 4 to 8.0 


29 to 70 


60° F. 


\ 


.076 


1.5 to 10.2 


35 to 100 


100 


1 


.084 


1.3 to 10.8 


10 to 50 


80 


2 


.080 


2.0to 10.6 


5 to 28 


80 


3 


.072 


4 


12 to 100 


60 


4 


.066 


7 


28 


35 


5 


.057 


5 


30 


86 


6 


.066 


4.5 


33 


70 


8 


.061 


4.5 


20 


70 


12 


.047 


7.5 


20 







An examination of the data in Appendix B shows that the 
coefficient c is independent of r and of s. If it is affected by 
the temperature it cannot be detected in these data. In 
relation to the diameters c evidently increases as the diam- 
eter decreases. A plot of diameters and c on coordinate 
paper gives a straight line and reveals the relation c = 
.0866 — .0033 d as most nearly averaging the results. This 
gives the following values for c: 



Diameters 


h 1 1 


1* 


2 


2i 1 3 


4 


5 16 18 


10 


12 


Coefficients 


.085 .083 


.081 


.080 


.079|.078 


.073 


.070) .067 .060 


.053 


.047 



Formulas (20) and (21) would be theoretically a little more 
accurate if v a were expressed in terms of the actual weight 
of air passing. This would involve the observed free air 
pressure and temperature at the time considered. Such a 



36 COMPRESSED AIR 

modification renders the formula much more laborious 
and would probably add nothing to its value for practical 
purposes. 

Table IX and Plates 0, I, II, III, and IV are based on 
formula (20). 

Art. 24. Theoretically Correct Friction Formula. 

The theoretically correct formula for friction in air 
pipes must involve the work done in expansion while the 
pressure is dropping. 

Let pi and p 2 be the absolute pressures at entrance and 
discharge of the pipe respectively and let Q be the total 
weight of air passing per second. 

Then the total energy in the air at entrance is 

i Pi i Qsi 2 

p a V a log ^ + J ^" 

and at discharge the energy is 

Va Va log ** + &* . 

Pa 2g 
The difference in these two values must have been absorbed 
in friction in the pipe. Hence, using the expression for 
work done in friction that was derived in Art. 23, we get 

^ IkT* = p a Va Aog 21 - log &) + £- fe 2 - S& . 
1^ \ Pa Pal & Q 

Numerical computations will show the last term, viz. 
— (s 2 2 — Si 2 ) is relatively so small that it can be neglected in 

any case in practice without appreciable error. Hence by a 
simple reduction we get 

log« £l = T f L X^ but *__S£_„, 

p 2 12 p a v a 4 X 144 

which when substituted gives 

! pi 4 X 144 k „ I 2 
p 2 12 p a a 

or considering p a as constant, 

logio — = Ci-s 2 
P2 d 

or logio Pi = logio Pi - ci -s 2 . (22) 

a 



FRICTION IN AIR PIPES 37 

In Eq. (22) Ci is the experimental coefficient and includes 
all constants, s is the velocity in the air pipe and varies 
slightly increasing as the pressure drops. All efforts so far 
have failed to get a formula in satisfactory shape that makes 
allowance for the variation in s. 

In working out Ci from experimental data s should be the 
mean between the Si and s 2 , and when using the formula 
the s may be taken as about 5 per cent greater than s x . 

Note that in the solution of Eq. (22) common logarithms 
should be used for convenience, allowing the modulus, 2.3+, 
to go into the constant C\. 

The working formula may be put in a different and 
possibly a more convenient form, thus. In the expression 

t Vi irk vy dl o 
loge - = T7> X ■ rs 3 

p 2 12 VaVa 

substitute for s its value 

_ 4 X 144 v a 



ird 2 r 



and reduce and we get 



Iv 2 
log p 2 = log pi - c 2 — - f- ■ . (23) 

Still another form is gotten thus. The whole weight of air 
passing is v a X w a = Q, and by Eq. (12) 

Q = v a -^- and therefore v a = 53 17 tQ . 
53.17 2 p a 

Pa 



Also r x = 22 an d Wa== _ 

p a 53.17 2 

Substitute these in (23) and it reduces to 

logP2 = logPl _^gy (24) 

In ordinary practice — may be taken as constant. If this 
be done Eq. (24) becomes 

log p 2 = log pi ~ c 3 t 5 ( — j * (24a) 

If t a = 525 and w a = .075, then c 3 = 7000 Cg. 



38 COMPRESSED AIR 

In (24) and (24a) p x varies between pi and p 2 . Careful 
computations by sections of a long pipe show p x to vary as 
ordinates to a straight line. Modifying the formulas to 
allow for this variation renders them unmanageable. In 
working out the coefficient p x may be taken as a mean 
between pi and p 2 , and in using the formula p may be taken 
as pi less half of the assumed loss Of pressure. 

As before suggested, common logarithms should be used in 
all the equations of this article. 

Finally it should be said that extreme refinement in com- 
puting friction in air pipes is a waste of labor, for there are 
too many variables in practical conditions to warrant much 
effort at precision. 

A study of the data collected in Appendix B gives 
values for c 2 , Eq. (24), that, for pipes three to twelve inches 
diameter, conform closely to the expression 

c 2 = .0124 - .0004 d, 

which gives the following: 

d" = 3 4 5 6 8 10 12 

~C 2 = .0112 .0108 .0104 .0100 .0092 .0084 .0080 
C 3 = 78.4 75.6 72.8 70.0 64.4 58.8 56.0 

With these coefficients p x in equations (24) and (24a) is to 
be taken in pounds per square inch. 

Equations (24) and (24a) are theoretically more correct 
than Eq. (20) and the coefficients of the former will not vary 
so much as those for the latter, but when the coefficients are 
correctly determined for Eq. (20) it is much easier to com- 
pute and can be adapted to tabulation, while Eq. (24) can- 
not be tabulated in any simple way. 

Example 24a. Apply formulas (20) and (24) to find the 
pressure lost in 1000' of 4" pipe when transmitting 1200 
cu. ft. free air per minute compressed to 150 gage when at- 
mospheric conditions are p a = 14.0, w a = . 073 and t a = 540. 

Solution by Eq. (20) : r = 150 + 14 = 11.71. By Table IX 

divide 23.44 by 11.71 and the result, 2 pounds, is the pres~ 
sure lost per 1000'. 



FRICTION IN AIR PIPES 



39 



Solution ofEq. (24) : The coefficient for a 4" pipe is .0108, 
and log pi = log (150 + 14) = 2.214844. 

Then log P2 = 2.214844 - .0108 jg ? X ^ (IgP x M *)* . 

The log of the last term is 3.791193 and its corresponding 
number is .006183. 

2.214844 - .006183 = 2.208661 = log p 2 . 
Whence p 2 = 161.7+ and pi — p 2 = 2.3. 

Art. 25. Efficiency of Power Transmission by Compressed 
Air. 

In the study of propositions to transmit power by piping 
compressed air, persons unfamiliar with the technicalities 
of compressed air are apt to make the error of assuming 
that the loss of power is proportional to the loss of pressure, 
as is the case in transmitting power by piping water. Fol- 
lowing is the mathematical presentment of the subject: 

pi = absolute air pressure at entrance to transmission pipe, 
p 2 = absolute air pressure at end of transmission pipe, 
Vi = volume of compressed air entering pipe at pressure pi, 
v 2 = volume of compressed air discharged from pipe at 
pressure p 2 . 

Then crediting the air with all the energy it can de- 
velop in isothermal expansion, the energy at entrance 



is piVi log 



2l = 

pa 



PiVi log n, and at discharge the energy is 



p 2 v 2 log^ 2 - = p 2 v 2 l°g r 2. 
Pi 

Hence efficiency E = ^ }°& " 2 = ^ 2 . 

PiVi l0g e 7*1 loge 7"i 



(25) 



Common logs may be used since the modulus cancels. The 
varying efficiencies are illustrated by the following tables. 

p a =14.5. pi = 87. ri = 6. log n = .7781. 



V-2 


85 


80 


75 


70 


65 


60 


r 2 


5.86 


5.52 


5.17 


4.83 


4.48 


4.14 


logr 2 . . 


.7679 


.7419 


.7135 


.6839 


.6513 


.6170 


E 


.987 


.953 


.917 


.879 


.837 


.793 



40 COMPRESSED AIR 

p a = 14.5. pi = 145. n = 10. log n- 1.000. 



V? •• 
r, .. . 
logr, 

E ... 



140 


135 


130 


125 


120 


9.66 


9.31 


8.97 


8.62 


8.28 


.9850 


.9689 


.9528 


.9355 


.9185 


.98 


.97 


.95 


.93 


.92 



The above examples illustrate the advantage in trans- 
mitting at high pressure. Of course the work cannot be 
fully recovered in either case without expanding down to 
atmospheric pressure, and to do this in practice heating 
would be necessary. It should be understood also that by 
reheating this efficiency can be exceeded. 

It should be noted also that the above does not apply 
in cases where the air is applied without expansion. In 
such cases the efficiency of transmission alone would be 

E= (P2 ~ Pa) V2 = n (f 2 - 1) m 

' (Pi - Pa) vi r 2 (n - 1) 
Example 25a. What diameter of pipe will transmit 5000 
cu. ft. of free air per minute compressed to 100 lbs. gage, with 
a loss of 10 per cent of its energy in 2500 feet of pipe, assum- 
ing p a — 14.0? 

Solution. 
«-m- 8.15; then by E q . (25) ^ = J&. 

Whence log r 2 = 0.8200; r 2 = 6.6, and 6.6 X 14 = 92.4. 
/ = 114 — 92.4 = 21.6 = loss of pressure. 

By Eq. (21), 

log d = |[log (.06X 2500) x(^P) 2 -log (2I.6X U^* 

= .7602, whence d = 5.75"., 

21 6 
Otherwise go into Table IX with loss for 1000 ft. = — ~ 

2.5 

= 8.64, and 8.64 X r = 8.64 X 7.37 = 63, (7.37 being the 
mean r) . Then opposite 5000 in the first column find nearest 
value to 63, which is 55 in the 6" column; showing the re- 
quired pipe to be a little less than 6". 



CHAPTER IV 

Other Air Compressors 

Art. 26. Hydraulic Air Compressors. — Displacement Type. 

Compressors of this type are of limited capacity and low 
efficiency, as will be shown. They are therefore of little 
practical importance. However, since they are frequently 
the subject of patents and special forms are on the market, 
their limitations are here shown for the benefit of those who 
may be interested. 

Omitting all reference to the special mechanisms by which 
the valves are operated, the scheme for such compressors is 
to admit water under pressure into a tank in which air has 
been trapped by the valve mechanisms. The entering 
water brings the air to a pressure equal to that of the water; 
after which the air is discharged to the receiver, or point of 
use. When the air is all out the tank is full of water, at 
which time the water discharge valves open, allowing the 
water to escape and free air to enter the tank again, after 
which the operation is repeated. Usually these operations 
are automatic. The efficiency of such compression is limited 
by the following conditions. 

Let P = pressure of water above atmosphere, or ordinary 
gage pressure, 
V = volume of the tank. 

Then P + p a = absolute pressure of air when compressed. 
The energy represented by one tank full of water is PV and 
by one tank full of free air when compressed to P + p a is 
P -\- v 

PaV l0g e ^ Fa = p a V log e T. 
Pa 

Therefore the limit of the efficiency is 

F _ p a V \0g e T _ p a l0g e T 
PV P 

41 



42 COMPRESSED AIR 

But P — pi — p a , where pi is the absolute pressure of the 
compressed air. Inserting this and dividing by p a the expres- 
sion becomes 

E = log e r = logi r X 2.3 > ^s 

r — 1 r — 1 

Table VII is made up from formula (26). 

The practical necessity of low velocities for the water 
entering and leaving the tanks renders the capacity of such 
compressors low in addition to their low efficiency. 

Should the problem arise of designing a large compressor 
of this class an interesting problem would involve the time of 
filling and emptying the tank under the varying pressure and 
head. Since it is not likely to arise space is not given it. 

It is possible to increase the efficiency of this style of 
compressor by carrying air into the tank with the water by 
induced current or Sprengle pump action — a well-known 
principle in hydraulics. At the beginning of the action 
water is entering the tank under full head with no resist- 
ance, and certainly additional air could be taken in with the 
water. 

Art. 27. Hydraulic Air Compressors. — Entanglement Type. 

A few compressors of this type have been built compara- 
tively recently and have proven remarkably successful as 
regards efficiency and economy of operation, but they are 
limited to localities where a waterfall is available, and the 
first cost of installation is high. 

; The principle involved is simply the reverse of the air-lift 
pump, and what theory can be applied will be found in 
Art. 33 on air-lift pumps. 

Fig. 8 illustrates the elements of a hydraulic air compressor. 
h is the effective water fall. 
H is the water head producing the pressure in the compressed 

air. 
t is a steel tube down which the water flows. 
S is a shaft in the rock to contain the tube t and allow the 
p water to return. 

R is an air-tight hood or dome, either of metal or of natural 
rock, in which the air has time to separate from the water. 



OTHER AIR COMPRESSORS 



43 



A is the air pipe conducting the compressed air to point 

of use. t 

b is a number of small tubes open B wP 

at top and terminating in a 

throat or contraction, in the 

tube t. 

By a well-known hydraulic prin- 
ciple, when water flows freely down 
the tube t there will occur suction in 
the contraction. This draws air 
in through the tubes b, which air 
becomes entangled in the passing 
water in a myriad of small bub- 
bles; these are swept down with 
the current and finally liberated 
under the dome R, whence the air 
pipe A conducts it away as com- 
pressed air. 

The variables involved practically 
defy algebraic manipulation, so that 
clear comprehension of the prin- 
ciples involved must be the guide 
to the proportions. 

Attention to the following facts is essential to an intelli- 
gent design of such a compressor. 

1. Air must be admitted freely — all that the water can 
entangle. 

2. The bubbles must be as small as possible. 

3. The velocity of the descending water in the tube t. 
should be eight or ten times as great as the relative ascend- 
ing velocity of the bubble. 

The ascending velocity of the bubble relative to the water 
increases with the volume of the bubble, and therefore 
varies throughout the length of the tube, the volume of 
any one bubble being smaller at the bottom of the tube 
than at the top. For this reason it would be consistent to 
make the lower end of the tube t smaller than the top. 




Fig 8. 



44 COMPRESSED AIR 

Efficiencies as high as 80 per cent are claimed for some 
of these compressors, which is a result hardly to have been 
expected. 

The great advantage of this method of air compression 
lies in its low cost of operation and in its continuity. Me- 
chanical compressors operated by the water power could be 
built for less first cost and probably with as high efficiency, 
but cost of operation would be much higher. 

Art. 28. Centrifugal Air Compressors. 

With the perfection of the steam turbine it has become 
practicable to deliver air at several atmospheres pressure 
through centrifugal machines. Such machines are not yet 
common, but doubtless in a few years they will become the 
standard machine where large volumes of air are needed at 
low and constant pressure. The simplicity, compactness and 
low first cost of such machines assure them a popularity. 

The theory of centrifugal fans or air compressors would 
involve matter not appropriate to the purpose of this vol- 
ume and is therefore omitted. 

In testing centrifugal compressors or blowers the orifice 
measurement, Art. 20, is the only practicable scheme. If 
the coefficients have not been determined for orifices suffi- 
ciently large to pass the volume of air, then more than one 
orifice can be placed in the orifice box. It is not necessary 
of course that these orifices all be of one size. 

The volume of air delivered and the efficiency of centrif- 
ugal fans and blowers is a matter little understood, seldom 
known, and often far from what is assumed or claimed. 
The remedy for this is to be found in intelligent use of the 
orifice, large and small; and for such purposes the deter- 
mination of orifice coefficients such as shown in Table V 
should be extended to orifices all the way up to two feet in 
diameter in order to test very large ventilating fans. 

Some theoretic discussion of centrifugal fans can be found 
in Trans. Am. So. C. E., Vol. 51, p. 12. See also "Turbo 
Compressors," Compressed Air, June, 1909, p. 5364, and En- 
gineering Magazine, Vol. 39, p. 237. 



CHAPTER V 

Special Applications of Compressed Air 

In this chapter attention is given only to those applications 
of compressed air that involve technicalities — with which 
the designer or user may not be familiar, or by the discussion 
of which misuse of compressed air may be discouraged and a 
proper use encouraged. 

Art. 29. The Return-Air System. 

In the effort to economize in the use of compressed air by 
working it expansively in a cylinder the designer meets 
two difficulties: first, the machine is much enlarged when 
proportioned for expansion; second, it is considerably more 
complicated; and third, unless reheating is applied the ex- 
pansion is limited by danger of freezing. 

To avoid these difficulties it has been proposed to use the 
air at a high initial pressure, apply it in the engine without ex- 
pansion, and exhaust it into a pipe, returning it to the intake 
of the compressor with say half of its initial pressure remain- 
ing. The diagram Fig. 9 will assist in comprehending the 
system. 

To illustrate the operation and theoretic advantages of 
the system assume the compressor to discharge air at 200 
pounds pressure and receive it back through R at 100 
pounds. Then the ratio of compression is only 2 and yet 
the effective pressure in the engine is 100 pounds. 

Evidently then with a ratio of compression and expansion 
of only 2 the trouble and loss due to heating are practically 
removed; and the efficiency in the engine even without a 
cut-off would be, by Eq. (15) 72 per cent. By the above dis- 
cussion the advantages of the system are apparent, and where 
a compressor is to run a single machine, as for instance a 
pump, the advantage of this return-air system will surely 

45 



46 



COMPRESSED AIR 



outweigh the disadvantage of two pipes and the high pres- 
sure, but where one compressor installation is to serve 
various purposes such as rock drills, pumps, machine shops, 
etc., the system cannot be applied. There should be a 
receiver on each air pipe near the compressor. 




Fig. 9. 



Engine 



Art. 30. The Return- Air Pumping System. 

Following the preceding article it is appropriate to men- 
tion the return-air pumping system. The economic principle 
involved is different from that of the return-air system in 
general. 

The scheme is illustrated in Fig. 10. It consists of two 
tanks near the source of water supply. Each tank is con- 
nected with the compressor by a single air pipe, but the air 
pipes pass through a switch whereby the connection with 
the discharge and intake of the compressor can be reversed, 
as is apparent on the diagram. In operation, the compressor 



SPECIAL APPLICATIONS OF COMPRESSED AIR 47 

discharges air into one tank, thereby forcing the water out 
while it is exhausting the air from the other tanks, thereby 
drawing the water in. The charge of air will adjust itself 
so that when one tank is emptied the other will be filled, 
at which time the switch will automatically reverse the 
operation. 




The economic advantage of the system lies in the fact that 
the expansive energy in the air is not lost as in the ordinary 
displacement pump (Art. 31). The compressor takes in air 
at varying degrees of compression while it is exhausting the 
tank. 

The mathematical theory and derivation of formulas for 
proportioning this style of pump are quite complicated but 
interesting. Since the system is patented, further discus- 
sion would seem out of place. It will be found in Trans. 
Am. So. C. E., Vol. 54, p. 19. 



48 COMPRESSED AIR 

Art. 31. Simple Displacement Pump. First known as 
the Shone ejector pump. 

In this style of pump the tank is submerged so that when 
the air escapes it will fill by gravity. The operation is simply 
to force in air and drive the water out. When the tank is 
emptied of water, a float mechanism closes the compressed- 
air inlet and opens to the atmosphere an outlet through 
which the air escapes, allowing the tank to refill. Various 
mechanisms are in use to control the air valve automati- 
cally. The chief troubles are the unreliable nature of float 
mechanisms and the liability to freezing caused by the 
expansion of the escaping air. Some of the late designs 
seem reliable. 

The limit of efficiency of this pump is given by formula 15 
and Table VI. The pump is well adapted to many cases 
where pumping is necessary under low lifts. In case of drain- 
age of shallow mines and quarries, lifting sewerage, and the 
like, one compressor can operate a number of pumps placed 
where convenient ; and each pump will automatically stop 
when the tank, is uncovered and start again when the tank is 
again submerged, 



CHAPTER VI 
The Air-lift Pump 

Art. 32. The air-lift pump was introduced in a practical 
way about 1891, though it had been known previously, as 
revealed by records of the Patent Office. The first effort at 
mathematical analysis appeared in the Journal of the Frank- 
lin Institute in July, 1895, with some notes on patent claims. 
In 1891 the United States Patent Office twice rejected an 
application for a patent to cover the pump on the ground 
that it was contrary to the law of physics and therefore would 
not work. Altogether the discovery of the air-lift pump 
served to show that at that late date all the tricks of air 
and water had not been found out. 

The air lift is an important addition to the resources of the 
hydraulic engineer. By it a greater quantity of water can 
be gotten out of a small deep well than by any other known 
means, and it is free from the vexatious and expensive depre- 
ciation and breaks incident to other deep well pumps. While 
the efficiency of the air lift is low it is, when properly pro- 
portioned, probably better than would be gotten by any 
other pump doing the same service. 

The industrial importance of this pump; the difficulty 
surrounding its theoretic analysis; the diversity in practice 
and results; the scarcity of literature on the subject; and the 
fact that no patent covers the air lift in its best form, seem 
to justify the author in giving it relatively more discussion 
than is given on some better understood applications of 
compressed air. 

Art. 33. Theory of the Air-lift Pump. 

An attempt at rational analysis of this pump reveals so 
many variables, and some of them uncontrollable, that 
there seems little hope that a satisfactory rational formula 

49 



50 



COMPRESSED AIR 



(7* 



will ever be worked out. However, a study of the theory- 
will reveal tendencies and better enable the experimenter to 
interpret results. 

In Fig. 11, P is the water discharge or reduction pipe with 
area a, open at both ends and dipped into the water. A 
is the air pipe through which air is forced into 
the pipe, P, under pressure necessary to 
overcome the head D. b is a bubble liberated 
in the water and having a volume which 
increases as the bubble approaches the top of 
the pipe. 

The motive force operating the pump is the 
buoyancy of the bubble of air, but its buoy- 
ancy causes it to slip through the water with 
a relative velocity u. 

In one second of time a volume of water 
= au will have passed from above the bubble 
to below it and in so doing must have taken 
some absolute velocity s in passing the con- 
tracted section around the bubble. 

Equating the work done by the buoyancy 
of the bubble in ascending, to the kinetic energy given the 
water descending we have 



D 




_.L — qw 



Fig. 11. 



wOu = wau — where w = weight of water, 



2g 



or 





a 



2<7 



(a) 



s 

— is the equivalent of the head h at top of the pipe which 

2 <7 ^ 


is necessary to produce s, therefore h = — • 

a 

Suppose the volume of air, 0, to be divided into an infinite 
number of small particles of air, then the volume of a particle 
divided by a would be zero and therefore s would be zero; 
but the sum of the volumes ( = 0) would reduce the specific 
gravity of the water, and to have a balance of pressure be- 
tween the columns inside and outside the pipe the equation 



wO = wah must hold. 



THE AIR-LIFT PUMPS 51 

Hence again h = — , showing that the head h depends only 
a 

on the volume of air in the pipe and not on the manner of 

its subdivision. 

The slip, u, of the air relative to the water constitutes 

the chief loss of energy in the air lift. To find this apply the 

law of physics, that forces are proportional to the velocities 

they can produce in a given mass in a given time. The 

force of buoyancy wO f of the bubble causes in one second a 

downward velocity s in a weight of water wau. Therefore 

wO s_ 
wau g 

a s 2 

Whence u = — U- . But — = — as proved above. 
as a 2 g 

Therefore M = i=y/2fi. (b) 

2 Y a 2 

This shows that the slip varies with the square root of 
the volume of the bubble. It is therefore desirable to 
reduce the size of the bubbles by any means found possible. 

If u = - , then the bubble will occupy half the cross 

section of the pipe. This conclusion is modified by the 
effect of surface tension, which tends to contract the bubble 
into a sphere. The law and effect of this surface tension 
cannot be formulated nor can the volume of the bubbles be 
entirely controlled. Unfortunately, since the larger bubbles 
slip through the water faster than the small ones, they tend 
to coalesce; and while the conclusions reached above may 
approximately exist about the lower end of an air lift, in 
the upper portion, where the air has about regained its 
free volume, no such decorous proceeding exists, but instead 
there is a succession of more or less violent rushes of air 
and foamy water. 

The losses of energy in the air lift are due chiefly to two 
causes: first, the slip of the bubbles, through the water, 
and second, the friction of the water on the sides of the 
pipe. As one of these decreases the other increases, for by 



52 



COMPRESSED AIR 



reducing the velocity of the water the bubble remains 
longer in the pipe and has more time to slip. 

The best proportion for an air lift is that which makes the 
sum of these two losses a minimum. Only experiment 
can determine what this best proportion is. It will be 
affected somewhat by the average volume of the bubbles. 
As before said, any means of reducing this volume will 
improve the results. 

Art. 34. Design of Air-lift Pumps. 

The variables involved in proportioning an air-lift pump 

are : — 

Q = volume of water to be lifted, 

h = effective lift from free water surface 
outside the discharge pipe, 

I = D + h = total length of water pipe 

above air inlet, ^^ 

D = Depth of submergence = depth at which 
air is liberated in water pipe meas- 
ured from free water surface outside 
the discharge pipe. 

v a = volume per second of free air forced into 
well, 

a = area of water pipe, 

A = area of air pipe, 

= volume of the individual bubbles. 



^f 



rH 



fF^ 



Air 



fr 



w 



&6. 



y /\\\ 



i 



4 
J 



f 



r 

h 



r 



^ 



The designer can control A, a, D + h and 
v a but he has little control over 0, and cannot 
foretell what D and Q will be until the pump 
is in and tested. 

When the pump is put in operation the 
free water surface in the well will always drop. 
What this drop will be depends first on the 
geology and second on the amount, Q, of water taken out. 
In very favorable conditions, as in cavernous limestone, 
very porous sandstone or gravel, the drop may be only a 
few feet, but in other cases it may be so much as to prove 
the well worthless. In any case it can be determined by 
noting the drop in the air pressure when the water begins 



Fig. 12. 



THE AIR-LIFT PUMPS 53 

flowing. If this drop is p pounds, the drop of water surface 
in the well is 2.3 X p feet. 

Unless other and similar wells in the locality have been 
tested, the designer should not expect to get the best pro- 
portion with the first set of piping, and an inefficient set of 
piping should not be left in the well. 

The following suggestions for proportioning air lifts have 
proved safe in practice, but, of course, are subject to revision 
as further experimental data are obtained. (See Figs. 13 
and 14.) 

Air Pipe. Since in the usually very limited space high 
velocities must be permitted we may allow a velocity of 
about 30 ft. per second in the air pipe. 

Submerqence. The ratio — is defined as the Sub- 

y D + h 

mergence ratio. Experience seems to indicate that this 

should be not less than one-half; and about 60 per cent 

is a common rule of practice. Probably the efficiency will 

increase with the submergence. The cost of the extra depth 

of well necessary to get this submergence is the most serious 

handicap to the air-lift pump. 

Ratio ^~- 

Let D = depth of submergence and h = effective lift — 
nD. Then the energy in the compressed air is 

, ID + 33.3\ D + 33.3 , . . , , . , 
P a v a log e f — ^- - — J) ~^y~ bem § the ratl ° of compres- 

sion, = r, and the effective work in water lifted is 

wQh = 62.5 QnD. 
If E be the efficiency of the system, then 

62.5 XQXnD = EX 2100 va X 2.3 logi (r), 
cubic foot units being used and common logs. 

Whence £ = -f- \ -^— (27) 

Q 77.3 E logi r 

Several apparently well proportioned wells are on record, 
see Art. 37, in which D is from 350 to 500 feet, n about § 



\ 



54 



COMPRESSED AIR 



and E 40 to 50 per cent. Taking n = § and E = 45 per 
cent, Eq. (27) reduces to 

Va = D 

Q 50 logio r 
From which the following table is computed. 



(27a) 



h 


D 


I 


Va 

Q 


h 


' D 


I 


V a 

Q 










167 


250 


417 


5.4 


6.6 


10 


16.6 


1.8 


200 


' 300 


500 


6.1 


33. 


50 


83.3 


2.5 


233 


350 


583 


6.6 


66. 


100 


166.0 


3.4 


267 


400 


667 


7.2 


100. 


150 


250.0 


4.1 


300 


450 


750 


7.8 


133. 


200 


333. 


4.8 


333 


500 


833 


8.4 










366 


550 


916 


8.9 











This table is reproduced in the curve plate V. It should 
be used only with full recognition of the assumptions on 
which it is based, and with due regard to what follows 
about velocities in the water pipe. The table has-been 
verified for h between 200 and 400 feet. For lower lifts it 
would be expected that a better efficiency could be obtained — 
the best data that can be found seem to indicate that such 
is the case. In consideration of this the dotted line on 
plate I may be a better guide than the full line. 

Velocity in the Water Pipe. 

This is the factor that most affects the efficiency, but un- 
fortunately, owing to the usual small area in the well, the 
velocity cannot always be kept within the limits desired. 
The complicated action and varying conditions in a well 
make the designer entirely dependent on the results of 
experience in fixing the allowable velocities in the discharge 
pipes. 

The velocity of the ascending column of mixed air and 
water should certainly be not less than twice the velocity at 
which the bubble would ascend in still water. This would 
probably put the most advantageous least velocity in any air 
lift at between five and ten feet, and this would occur where 
the air enters the discharge pipe. 



THE AIR-LIFT PUMPS 



55 



■■ ■■ 


■■ i "" 






± ■ ::: ~__:_ _ 








----- ------- 






















_::i;:: i i 


















i\i ! | ■ i ■ , i 






_, ^^ 








ii - i cr 1 ^+3 - 




X r t "2 » 


x ! ' till ■ i Ml I 


± _ ._. ^ Oj) 








± . bto G.H 
















1 I Ml IVJ p <D V 




















l l_ |_} ~ ^ ± 


i ifJ \ < 1 ' I 1 ' M 


1 | a, w mw 




" O P U . 




' -l-X ± U +^> £ 




'Ml 1 o r 




1 ^ T3 °, 


r\Vi i 


Mi > a 


\\ 


1 S-l (|1 w 


i VI 








X'N ' iii 


___L ^ ^ 


1 M 1 1 M ■ II I Vl\ 




M i 1 , 1 \T I 


XX 


II III Ml \> ■ M 




1 II I J 1 1 1 1 1 1 \l \ II 1 1 1 1 1 1 1 


j 1 , 










1 j 1 i 1 '■ ' ' N \ M II 1 1 1 


I.I ' 








III 






III II III iVN ill 1 


1 Ii 






\ \ 










it — - " — :: "" — 






i ,I|IM 1 | M 1 I 1 :\ |\ | 




II 'Ml \l X 




i i ■ ' ■ ' ' i \ \ 




1 1 1 III MM hl\ f 


,i 




ST ' i ! I 1 1 I 




i i 1 i 1 Mi 




S m M M i i i i i , i 1 




w , ' ' 1 




\ \ 


1 i ' 




M 1 I M M . 1 1 


SJ i K I [ M ' ! 


1 1 1 1 1 1 M 1 M II II 




1 I 1 | 1 Mil 1 1 i ill i 


\ \ , i 














m | i i ii Mi 1 ! i 


1 Ml ■* 


, i ■ i 


1 m i X i v r 




npL v 


11 




i ii i • . i 1 i i ■ i i 1 










. Mm i ' i rNJ 1 \ M 




1 MM Ml | X | '■-■ | 




1 1 1 II j | | 1 1 MO VI l| 1 i 


h 1 M 1 II 1 i j 1 I 1 














x n . 




1 sis, 















o -d 

o — < 
■* cS 



H 



'A 



56 COMPRESSED AIR 

The velocity at any section of the pipe will be 



a 



where Q and v are the volumes of water and air respectively 
and a the effective area of the water pipe, v increases 
from bottom to top probably very nearly according to the 
formula 



I - | 



1 - - 1 



(28) 



where 



r = ratio of compression under running conditions, 
I = total length of discharge pipe above air inlet, 
x = distance down from top of discharge pipe to section 
where velocity is s. 

The formula (28) is based on the assumption that the vol- 
ume of air varies as the ordinate to a straight line while 
ascending the pipe through length I. As the volume of each 
bubble increases in ascending the pipe, the velocity of the 
mixture of water and air should also increase in order to 
keep the sum of losses due to slip of bubble and friction of 
water a minimum; but for deep wells with the resultant 
great expansion of air the velocity in the upper part of the 
pipe will be greater than desired, especially if the discharge 
pipe be of uniform diameter. Hence it will be advantageous 
to increase the diameter of the discharge pipe as it ascends. 
The highest velocity (at top) probably should never exceed 
twenty feet per second if good efficiency is the controlling 
object. 

Good results have been gotten in deep wells with velocities 
about six feet at air inlet and about twenty feet at top. (See 
Art. 37.) 

Fig. 13 shows the proportions and conditions in an air 
lift at Missouri School of Mines. 

The flaring inlet on the bottom should never be omitted. 
Well-informed students of hydraulics will see the reason for 
this, and the arguments will not be given here. 



THE AIR-LIFT PUMPS 57 

The numerous small perforations in the lower joint of the 
air pipe liberate the bubbles in small subdivisions and some 
advantage is certainly gotten thereby. 

No simpler or cheaper layout can be designed, and it has 
proved as effective as any. It is the author's opinion that 
nothing better has been found where submergence greater 
than 50 per cent can be had. 

Art. 35. The Air Lift as a Dredge Pump. 

The possibilities in the application of the air lift as a 
dredge pump do not seem to have been fully appreciated. 
This may be due to its being free from patents and therefore 
no one being financially interested in advocating its use. 

With compressed air available a very effective dredge 
can be rigged up at relatively very little cost and one that 
can be adapted to a greater variety of conditions than those 
in common use, as the following will show. 

Suggestions: 

Clamp the descending air pipe to the outside of the dis- 
charge pipe. Suspend the discharge pipe from a derrick 
and connect to the air supply with a flexible pipe (or hose). 
With such a rigging the lower end of the discharge pipe can 
be kept in contact with the material to be dredged by lower- 
ing from the derrick ; the point of operation can be quickly 
changed within the reach of the derrick, and the dredge can 
operate in very limited space. In dredging operations the 
lift of the material above the water surface is usually small, 
hence a good submergence would be available. The depth 
from which dredging could be done is limited only by the 
weight of pipe that can be handled. 

Art. 36. Testing Wells with the Air Lift. 

The air lift affords the most satisfactory means yet found 
for testing wells, even if it is not to be permanently installed. 
Such a test will reveal, in addition to the yield of water, the 
position of the free water surface in the well at every stage 
of the pumping, this being shown by the gage pressures. 
However, some precautions are necessary in order properly 



58 



COMPRESSED AIR 



to correct the gage readings for friction loss in the air 
pipe. 

The length of air pipe in the well and any necessary cor- 
rections to gage readings must be known. 

The following order of proceeding is recommended. 

At the start run the compressor very slowly and note the 
pressure pi at which the gage comes to a stand. This will 
indicate the submergence before pumping commences, since 
there will be practically no air friction and no water flowing 
at the point where air is discharged. Now suddenly speed 
up the compressor to its prescribed rate and again note the 
gage pressure p 2 before any discharge of water occurs. Then 
Pi — Pi = Vf i s ^ ne pressure lost in friction in the air pipe. 
When the well is in full flow the gage pressure p 3 indicates 
the submergence plus friction, or submergence pressure p s = 
p 3 — pf. The water head in feet may be taken as 2.3 X p. 
Then, knowing the length of air pipe, the distance down to 
water can be computed for conditions when not pumping 
and also while pumping. 

4 

Art. 37. Data on Operating Air Lifts. 

In Figs. 13 and 14 are shown the controlling numerical 
data of two air lifts at Rolla, Mo. These pumps are perhaps 
unusual in the combination of high lift and good efficiency. 
The data may assist in designing other pumps under some- 
what similar circumstances. 

The figures down the left side show the depth from sur- 
face. The lower standing-water surface is maintained 
while the pump is in operation; the upper where it is not 
working. 

The broken line on the right shows, by its ordinate, the 
varying velocities of mixed air and water as it ascends the 
pipe. 

The pump Fig. 13 delivers 120 gallons per minute with a 

ratio — — = 6.0. The submergence is 58 per cent and 

Water 

efficiency = Net energy in water lift _ 5Q cent 

pv log e r 



THE AIR-LIFT PUMPS 



59 



The pump Fig. 14 delivers 290 gallons per minute with 

a ratio ree air = 5.2. Submergence = 64 per cent and 
Water 

efficiency = Net energy in water lift = ^ ^ ^ 

pv log e r 



.rfF 2 *- 



.^= 3 



^s 




Fig. 13. 




Fig. 14. 



The volumes of air used in the above data are the actual 
volumes delivered by the compressor. The volumetric 
efficiencies of the compressors by careful tests proved to be 
about 72 per cent. 



CHAPTER VII 

Examples and Exercises 

Art. 38. The following combined example includes a solu- 
tion of many of the types of problems that arise in designing 
compressed-air plants. The student will find it well worth 
while to become familiar with every step and detail of the 
solutions, which are given more fully than would be nec- 
essary except for a first exercise. 

Example 26. An air-compressor plant is to be installed 
to operate a mine pump under the following specifications: 

1. Volume of water = 1500 gallons per minute. 

2. Net water lift = 430 feet. 

3. Length of water pipe = 1280 feet. 

4. Diameter of water pipe = 10 inches. 

5. Length of air pipe = 1160 feet. 

6. Atmospheric pressure = 14.0 pounds per sq. in. 

7. Atmospheric temperature 50° F. 

8. Loss in transmission through air line = 8 per cent of 
the pv log e r at compressor. 

9. Mechanical efficiency of the pump = 90 per cent as 
revealed by the indicators on the air end and the known 
work delivered to the water. 

10. Average piston speed of pump = 200 feet per minute. 

11. Mechanical efficiency of the air compressor = 85 per 
cent as revealed by the indicator cards. 

12. R.P.M. of air compressor = 90 and volumetric effi- 
ciency = 82 per cent. 

13. In compression and expansion n = 1.25. 

Preliminary to the study of the problems involving the air 
we must determine: 

(a) Total pressure head against which the pump must work. 

By the methods taught in hydraulics the friction head in a 
pipe 10 inches in diameter, 1280 feet long, delivering 1500 

60 



EXAMPLES AND EXERCISES 61 

gallons per minute, is about 2Q feet. Therefore the total 
head = 450 feet. 

(6) Total work (Wi) delivered to the water in one minute. 
Wi = 1500 X 8J X 450 = 5,625,000 foot-pounds. 

(c) Total work (W) required in air end of pump. 

By specification 9, W = ^ = 6,250,000 ft.-lbs. = 190 

horse power. 

For the purpose of comparison, two air plants will be 
designed; the first, designated d, as follows: 

(d) Compression single-stage to 80 pounds gage. No 
reheating. No expansion in air end of pump. Pump direct 
acting without fly wheels. 

Determine the following: 

(dl) Air pressure at pump and pressure lost in air pipe. 

By specification 8 and Eq. (25), 

lo — 

^ = — , or log &■ = .92 log 6.72. 

100 , 80 + 14' & 14 * 

l0g ^4~ 
Whence, using common logs, log '—■ = 0.76118 and 

p 2 = 80.78. 

Then lost pressure = pi - p 2 = 94 - 80.78 = 13.22 = f, 
and gage pressure at pump = 80 — 13.22 = 66.78. 

{62) Ratio between areas of air and water cylinders in pump. 

The pressure due to 450 feet head = 450 X .434 = 194.3, 

say 195 pounds, per sq. in.; and since pressure by area must 

i , ,i , i area air end 195 i 

be equal on the two ends, ■ = — — - = 3 nearly. 

area water end 66.78 

(dS) Volume of compressed air used in the pump. Cubic 
feet per minute: 

Evidently from solution (d2) the volume of compressed 
air used in the pump will be three times that of the water 
pumped, or 

v = — — X3 = 601.6 cu. ft. per min. 

7.48 



62 COMPRESSED AIR 

(d4) Diameters of air cylinder and of water cylinder. 

Since the piston speed is limited to 200 feet per min. 
(spec. 10) and the volume is 1500 gallons, we have, when all is 
reduced to inch units and letting a — area of water cylinder, 
a X 200 X 12 = 1500 X 231. Whence a = 144 sq. in. which 
requires a diameter of about 13f inches. 

The area of air cylinder is by d 2 three times that of the 
water cylinder, which gives a diameter 23 J inches for the air 
cylinder. 

(d5) Volume of free air. 

From dl, r at the pump = 5.76. Therefore 

v a = 601.6 X 5.76 = 3465 cu. ft. per min. 
(dQ) Diameter of air pipe. 
The mean r in the air pipe is — — — — : — = 6.24. Using 

this in Eq. (21) with c = .06, we get d = 5 inches. 

13 99 

Or using plate III with r X 13.22 -v- 1.160 or r X ^^ 
* F 1.160 

as vertical ordinate and 3465 as horizontal ordinate, the in- 
tersection falls near the 5-inch line. 

{dl) Horse power required in steam end of compressor. 
By table II the weight per foot of free air is .07422 pound 
per cu. ft. Total weight of air compressed = Q 

Q = .07422 X 3465 = 257 pounds per min. 

In table I opposite r = 6.72 in column 9 find by inter- 
polation .3736. Then 

Horse power = 2.57 X .3736 X (460 + 50) = 489.6 in air 

end = '— = 576 in steam end. 

.85 

The second plant will be designated by the letter e and 
will be two-stage compression to 200 pounds gage at air 
compressor, will be reheated to 300° at the pump and used 
expansively in the pump; the expansion to be such that the 
temperature will be 32° at end of stroke. 

(el) Air pressure at pump. 

Apply Eq. (25) as in dl. In this case n (at the com- 
pressor) = 15.3 and r 2 (at the pump) = 12.3. Therefore 



492 



EXAMPLES AND EXERCISES 63 

pressure at the pump = 12.3 X 14 = 172.3 and the lost 
pressure = 214 - 172.3 = 41.7 = f, 

(e2) Point of cut-off in air end of pump = fraction of 
stroke during which air is admitted. 

By Eq. (11) viz. - = ( -) , putting in numbers we get 
k W 

( - ) whence - 1 = .176, which is the point of cut-off, 
760 \v 2 J v 2 

and v 2 = 5.68 v±. 

Or go into table I in column 5, find the ratio — - = 1.545, 

and in same horizontal line in column 3 find .176. 

(e3) Volume of compressed hot air admitted to air end of 
pump. 

Apply Eq. (9) viz. Work = PlVl ~ ^ + p^ - p a v 2 . 

n — 1 

In this we have Work = 6,250,000, v 2 = 5.68 v h p x = 214, 
n — 1 = .25, p a = 14, and p 2 must be found by Eq. (11a), or 
it may be gotten from table I by noting that for a tempera- 
ture ratio of 1.545 the pressure ratio is 8.8 and - = .1136, 

r 

therefore p 2 = .1136 X 172.3 = 19.57. This would give gage 
pressure = 5.57. ' 

Inserting these numerals in Eq. (9) we get 

6,250,000 = 144 Vl / 172 -3~5.68X19.57 + 172.3- 14 X5.68Y 

Whence Vi = 128.6 cu. ft. per min. 

(e4) Diameter of air cylinder of pump when air and water 
pistons are direct connected. 

Since expansion ratio is 5.68 (see e2) and the volume before 
cut-off is 128.6, the total piston displacement is 128.6 X 5.68 = 
730.8 cu. ft. per min. When the air and water pistons are 
direct connected they must travel through equal distances, 
therefore the air piston travels through 200 ft. per min. (spec. 
10). Then if a = area of piston in sq. ft. we have 

200 a = 730.8 and a = 3.654 sq. ft. 

By table X the diameter is 26 inches nearly. 



64 COMPRESSED AIR 

(e5) Volume of cool compressed air used by pump, cu. ft. 
per min. 

By e3 the volume of hot compressed air is 128.6, and since 
under constant pressure volumes are proportional to abso- 
lute temperatures, we have 

= - — Whence v = 86.3 cu. ft. per min. . 

128.6 760 P 

(e6) Volume of free air used. 

From el the ratio of compression at the pump is 12.3 and 
from e5 the volume of cool compressed air is 86.3, therefore 

the volume of free air is 86.3 X 12.3 = 1061.6. 

■- 

{el) Diameter of air pipe. 

The r for Eq. (21) is 12 - 3 + 15 - 3 = 13.8. 

Applying Eq. (21) with coefficient c — .07 we have 

/.07 X 1160 X (^)Y 

d = \ V b0 ' ) =2.13 inches. 

\ 41.7 X 13.8 / 

(e8) Horse power required in steam end of compressor. 

By dl the weight per cu. ft. of free air is .07422 and by e6 
the volume of free air compressed is 1061.6. Therefore the 
total weight compressed is .07422 X 1061.6 = 78.8 pounds 
per min., and the initial absolute temperature is 510. 

In the two-stage compression r 2 = 15.3, and assuming equal 

work in the two stages the n = Vl5.3 = 3.91 nearly. 

(See Art. 12.) Then going into Table I with r = 3.91 in column 

9 find .2525. Hence horse power = .2525 X 78.8 X 510 = 

101.5 for one stage, and for the two stages 101.5 X 2 = 203, 

203 
and (spec. 11) = 238.8 horse power in steam end. 

.85 

(e9) Diameter of air compressor cylinders, assuming 3- 
foot strokes and 2\-inch piston rods, equal work in the two 
cylinders and allowing for volumetric efficiency. 

By e6 the free air volume is 1061.6 and (spec. 12) the 
volumetric efficiency = 82 per cent. Therefore the piston 

displacement = — — t— — 1294.6 cu. ft. per min. 

.82 



EXAMPLES AND EXERCISES 65 

By spec. 12 the R.P.M. = 90. Therefore the displace- 
ment per revolution = 14.7, nearly, for the low-pressure 
cylinder. Add to this the volume of one piston rod length 
of 3 feet, which is 3 X .0341 = 0.1023. Whence the volume 
per revolution must be 14.8 or for one stroke 7.4. Whence 

7 4 
the area = - L — = 2.466 sq. ft. By Table X the diameter 

o 

is 21 J inches nearly for low-pressure cylinders. 

The high-pressure cylinder must take in the net volume 
of air compressed to r = 3.91 (see e8). Therefore the net 

volume per revolution = '- — - = 3.02. Add one piston 

F 90 X 3.91 F 

rod volume and get 3.12 per revolution or 1.56 per stroke 

and an area of 0.53 sq. ft. By Table X this requires a 

diameter of 10 inches nearly. 

(elO) Temperature of air at end of each compression stroke. 

In Table I the ratio of temperatures for r = 3.91 is 1.313. 
Hence the higher temperature = 510 X 1.313 = 669 absolute 
= 209 F. 

EXERCISES 

1 . (a) Assuming isothermal conditions, how many revo- 
lutions of a compressor 16" stroke, 14" diameter, double 
acting, would bring the pressure up to 100 lbs. gage in a 
tank 4 feet diam. X 12 feet length, atmospheric pres- 
sure = 14.5 per sq. in. ? 

(6) What would be the horse power of such a compressor 
running at 100 R.P.M. ? 

(c) What would be the horse power if the compression 
were adiabatic ? 

(d) What weight of air would be passed per minute when 
R.P.M. = 100 ? 

2. The air end of a pump (operated by compressed air) is 
20" diam. by 30" stroke, R.P.M. = 50, cut-off at \ stroke, 
free air pressure = 14.0, T a = 60°, compressed air delivered 
at 75 lbs. gage, T = 60° and n = 1.41. 

(a) Find work done in horse power. 

(b) Find weight handled per minute. 

(c) Find temperature of exhaust (degrees F). 



66 COMPRESSED AIR 

3. With atmospheric pressure, p a = 14.7, and T a = 50°, 
under perfect adiabatic compression, what would be the pres- 
sure (gage) and temperature (F.) when air is compressed to 

(a) \ its original volume ? 

(6) \ its original volume ? 

(c) \ its original volume ? 

(d) I its original volume ? 

(e) tV its original volume ? 

4. With p a = 14.1 and T a = 60° what will be the pres- 
sure of a pound of air when its volume = 3 cu. ft. ? 

5. What would be the theoretic horse power to compress 
10 pounds of air per minute from p a = 14.3 and T a = 60° to 
90 pounds gage ? 

(a) Compression isothermal. 
(6) Compression adiabatic. 

6. Find the point of cut-off when air is admitted to a motor 
at 250° F. and expanded adiabatically until the temperature 
falls to 32° F. 

7. What is the weight of 1 cu. ft. of air when p a = 14.0 
and T a = - 10° ? 

8. A compressor cylinder is 20" diam. by 26" stroke double 
acting. Clearance = 0.8%, piston rod = 2", R.P.M. = 100, 
atmospheric pressure, p a = 14.3, atmospheric temperature 
= T a = 60° F., and gage pressure = 98 lbs. 

Determine the following: 
(a) Compression isothermal. 

la. Volume of free air compressed, cu. ft. per min. 

2a. Volume of compressed air, cu. ft. per min. 

3a. Work of compression, ft.-lbs. per min. 

4a. Lbs. of "cooling water, T l = 50°, T 2 = 75°. 
(6) n = 1.25 and air heated to 100° while entering. 

16. Volume of free air compressed per min. 

26. Volume of cool compressed air per min. 

36. Work done in compression. 

46. Temperature of air at discharge. 

9. The cylinder of a compressed-air motor is 18" X 24", 
the R.P.M. = 90, air pressure 100 pounds gage. In the 



EXAMPLES AND EXERCISES 



67 



motor the air is expanded to four times its original volume 
(cut-off at J), with n = 1.25. 

(a). Determine the horse power and final temperature 
when initial T = 60° F. 

(b) . Determine the horse power and final temperature when 
initial T = 212° F. 

io. Observations on an air compressor show the intake 
temperature to be 60° F., the r = 7 and the discharge tem- 
perature = 300 F. What is the n during compression ? 

Hint. Use Eq. (11a) with n unknown. 

ii. In a compressed-air motor what percentage of power 
will be gained by heating the air before admission from 
60° to 300° F. ? 

12. If air is delivered into a motor at 60° F. and the ex- 
haust temperature is not to fall below 32° F., what ratio of 
expansion can be allowed ? What could be allowed if initial 
temperature were 300° ? What would be the ratio of work 
gotten in the two cases ? 

13. A compressed-air locomotive system is estimated to 
require 4000 cu. ft. per min. of free air compressed to 500 
pounds gage in three stages with complete cooling between 
stages. 

Assume n = 1.25, p a = 14~5, T a = 60°, Vol. Eff. = 80 per 
cent, Mechanical Eff. = 85 per cent and R.P.M. = 60. 

Compute the volume of piston stroke in each of the three 
cylinders and the total horse power required of the steam end. 

14. A compressor is guaranteed to deliver 4 cu. ft. of free 
air per revolution at a pressure of 116 (absolute). To test 
this the compressor is caused to deliver into a closed system 
consisting of a receiver, a pipe line and a tank. Observed 
conditions are as follows: 



Pressures at start (ab.) . . . 
Temperatures at start (F.) 
Pressures at end (ab.) 
Temperatures at end (F.) 
Volumes (cu. ft.) 



Receiver. 


Pipe. 


Tank. 


14.5 


14.5 


14.5 


60.0 


60.0 


60.0 


116.0 


116.0 


116.0 


150.0 


90.0 


60.0 


50.0 


10.0 


100.0 



68 COMPRESSED AIR 

How many revolutions of the compressor should produce 
this effect ? 

15. Find the discharge in pounds per minute through a 
standard orifice when d = 2", i = 5", t = 600° and p a = 
14.0. 

16. What diameter of orifice should be supplied to test 
the delivery of a compressor that is guaranteed to deliver 
1000 cu. ft. per min. of free air ? 

17. What is the efficiency of transmission when air pres- 
sure drops from 100 to 90 pounds (gage) in passing through 
a pipe system ? 

18. A compressor must deliver 100 cu. ft. per min. of com- 
pressed air at a pressure = 90 pounds, gage, at the terminus 
of a pipe 3000 ft. long and 3" diameter. p a = 14.4, T a = 
60° F. 

(a) Assuming a Vol. Eff. = 75 per cent, what must be the 
piston displacement of the compressor ? 

(b) What pressure is lost in transmission ? 

(c) What horse power is necessary in steam end of com- 
pressor if n = 1.25 and the mechanical efficiency = 85 per 
cent? 

(d) What would be the efficiency of the whole system 
if air is applied in the motor without expansion, the 
efficiency to be reckoned from steam engine to work done 
in motor ? 

19. It is proposed to convey compressed air into a mine 
a distance of 5000'. The question arises: Which is better, 
a3"ora4" pipe? 

Compare the propositions financially, using the following 
data: Nominal capacity of the plant = 1000 cu. ft. free 
air per min., Vol. Eff. of compressor = 80 per cent, n = 1.25 
gage pressure at compressor = 100, weight of free air w a = 
.074, p a = 14.36, weight of 3" pipe = 7.5 and of 4" pipe = 
10.7 pounds per foot. Cost of pipe in place = 4 cents per 
pound. Cost of one horse power in form of pv log r for .10 
hours per day for one year = $150. Plant runs 24 hours per 
day. Rate of interest = 6 per cent. 



EXAMPLES AND EXERCISES 69 

20. Air enters a 4" pipe with 60 feet velocity and 80 
pounds gage pressure; the air pipe is 1500 feet long. 

(a) Find the efficiency of transmission. 

(b) Find horse power delivered at end of pipe in form 
pv log r. 

(c) Find horse power delivered at end of pipe in form 

PgXV. 

21. An air pipe is to be 2000 feet long and must deliver 50 
horse power at the end with a loss of 5 per cent of the pv log r 
as measured at compressor. The pressure at compressor is 
75 pounds gage. p a = 14.7. Find diameter of pipe. 

22. Modify 21 to read: 50 horse power . . . with loss of 
5 per cent of the energy in form P g X v, where P g is gage 
pressure, and find diameter of air pipe. 

23. In case 21 let pressure at compressor be 250 pounds 
gage and find diameter of air pipe. 

24. The air cylinder of a compressed-air pump is 20 " 
diam. by 30" stroke. The machine is double acting and 
makes 50 R.P.M. The cut-off is to be so adjusted that the 
temperature of exhaust shall be 30°. p a = 14.5 and the r at 
pump = 8. 

(a) Find cut-off when initial temperature is 60° F. 
(6) Find cut-off when initial temperature is 250° F. 

(c) Find horse power in case (a) . 

(d) Find horse power in case (b). 

(e) In case (a) find efficiency in applying the pv log r of 
cool air. 

(/) In case (b) find efficiency in applying the pv log r of 
cool air. 

(g) Find the volumes of free air used in cases (a) and (b) . 

25. A compound mine pump is to receive air at 150 lbs; 
gage; this is to be reheated from 60° to 250° F., let into the 
H.P. cylinder of the pump and expanded until the temperature 
is 32°, then exhausted into an interheater where the tempera- 
ture is again brought to 250°. It then goes into the L.P. 
cylinder and is expanded down to atmospheric pressure 
= 14.5, (ab.). 

(a) Find point of cut-off in each cylinder, n = 1.25. 



TO COMPRESSED AIR 

(b) If the air is compressed in two stages with n = 1.25, 
what will be the efficiency of the system, neglecting friction 
losses ? 

(c) How much free air will be required to operate the 
pump if it is to deliver 250 horse power, assuming the efficiency 
of the pump to be 80 per cent reckoned from the work in the 
air end ? 

(d) If the pump strokes be 60 per min. and 26" long, fix 
diameters of cylinders in case (c). 

26. Compute the horse power of a motor passing one 
pound of air per minute admitted at 200° F. and 116.0 
pounds (ab.) r = 8, the air to be expanded until pressure 
drops to 29 pounds (ab.), r = 2. 

27. A pump to be operated by compressed air must deliver 
1000 gallons of water against a net head of 200' through 
800' of 10" pipe. The pump is double acting, 30" stroke, 
50 strokes per min. The air is reheated to 275° F. before 
entering the pump. The cut-off is so adjusted that with 
n = 1.25 the temperature at exhaust = 36° F. Mec. Effi. 
of pump = 80%. Air pressure at compressor = 90 pounds 
gage, p a = 14.4. Length of air pipe = 2000'. Permissible 
loss in transmission = 7 per cent of the pv log r at com- 
pressor. Mec. Effi. of compressor = 85 per cent. Vol. Effi. 
= 80 per cent. 

(a) Proportion the cylinders of the pump. 
(6) Determine the volume of free air used. 
(c) Determine the diameter of air pipe. 

28. Compare the volume displacement of two air com- 
pressors, one at sea level and the other at 12,000 feet eleva- 
tion; the compressors to handle the same weight of air. 

29. (a) An exhaust pump has an effective displacement of 
3 cu. ft. per revolution. How many revolutions will reduce 
the pressure in a gas tank from 30 to 5 pounds absolute, 
volume of tank = 400 cu. ft.? 

(6) If the pump is delivering the gas under a constant 
pressure of 30 pounds, what is the maximum rate of work 
done by the pump — foot pounds per revolution? 



PLATES a^d TABLES 



71 



NOTES ON TABLE I. 

The table is the solution of formulas n, na, 8a and ia. 

When the weight of air passed and its initial temperature are known, 
the table covers all conditions including elevation above sea level, reheating, 
and compounding. 

In compounding, either compression or expansion, the same weight 
goes through each cylinder. Then knowing the initial t and the r for each 
cylinder, find from the table the work done in each cylinder and add. Usu- 
ally the r and / are assumed the same for each cylinder — then take out the 
work for one stage and multiply by the number of stages. 

The table does not include friction in the machine nor the effect of clear- 
ance in expansion motors. 

The table is equally applicable to compression or expansion provided 
the correct r be taken in cases of expansion. 

Example. Air is received at such a pressure that r = 8. What should be 
the cut-off in order that the temperature drop from 6o° to 32 F. ? Expan- 
sion adiabatic. 

The ratio of temperatures is 1.057, which by linear interpolation corre- 
sponds to a volume ratio of .871 or cut-off at about |-. 

What would be the pressure at exhaust? 

The two ratios above correspond to a - = .825. Therefore the final 

pressure is .825 X initial pressure. 

To find the foot-pounds of work per hundred pounds of air compressed 
(or expanded) multiply the number opposite the r in column 7, 8 or 11 as 
the case may be by the absolute initial temperature, t. 

To find the weight compressed, go into Table II with known atmospheric 
conditions and the cubic feet capacity of the machine. 

To find the weight in case of expansion, take the number from Table II 
corresponding to the atmospheric conditions, multiply this by r X cubic 
feet capacity of the machine X cut-off fraction. 

To find the horse power per hundred pounds of air passed per minute, 
multiply the number opposite r in column 9, 10 or 12 as the case may be by 
the absolute temperature, /. 



72 



TABLE I. 



GENERAL TABLE RELATING TO AIR COM- 
PRESSION AND EXPANSION 















Work Factor 










Ratio of 




Work Factor. 


for Isothermal 


ti 




Ratio of 


Greater to 


Air Heated by Compression. 


Compression. 


o 


O 


Less to 


Less Tem- 











in jj 

0) O 


t-H O 

.b 


Greater 
Volume — 


perature. — 
Tempera- 


K=S3 


n 
17 


H.P. Fac- 


h, 




^ s 


S cc? 
O & 


% 1 


Air 


Hot. 


tures Ab- 
solute. 


v l n 


» — 1 
- 1 \ 


tor per 100 
Pounds per 


u . 


p, 2 

O <U 


^ w 


co 1 




T 




x - 


7T 


Minute 


1!. **i 3 

**• O 


-5 p. 


o ° 

a! 


^ 

■si 

P< 




ft 


n— 1 


\r -1/ 

Factor K for one 
pound. 


K 
330 


CO -t-> <u 
in a 

11 £ ° 


X 




n = 


n — 


n = 


n = 






n = 


n = 




K 














n= 1.25 


n= 1. 41 










r 




1.25 


1. 41 


1.25 


1. 41 




i- 2 5 


1 .41 




330 


I 


V2 


V% 


h 


h 


Ft. -Lbs. 


Ft.-Lbs. 


H.P. 


H.P. 


Ft.-Lbs. 


H.P. 




r 


Vi 


I. OOO 


h 


h 














i 


1 . 0000 


I. OOO 


1 .000 


1 .000 


0.0 


0.0 


.0 


.0 


0.0 


.0 


i . i 


.9091 


•927 


•935 


1. 019 


1.028 


5-i3* 


5-!40 


.0155 


•oi55 


5.068 


.OI53 


1.2 


• 8 333 


.862 


.877 


1.037 


1.054 


9.863 


9-932 


.0298 


•0301 


9.694 


•0293 


i-3 


.7692 


.812 


.830 


1.054 


1.079 


14.329 


14.450 


-0434 


•0437 


13-950 


.0422 


1.4 


•7*43 


' .7 6 4 


•787 


1.070 


1-103 


18.503 


18.766 


.0560 


.0568 


17.890 


.0542 


i. 5 


.6667 


•723 


•75o 


1.085 


1-125 


22.465 


22.827 


.0680 


.0691 


21.559 


•o653 


i.6 


.6250 


.687 


.717 


1. 100 


1. 146 


26.186 


26.704 


•o793 


.0809 


24.991 


•0757 


i-7 


.5882 


•654 


.686 


1 . 112 


1. 166 


29-775 


30.4I7 


.0902 


.0921 


28.214 


•0855 


i.8 


•5555 


.625 


•659 


1-125 


1. 186 


33-I78 


33.985 


.1005 


.1029 


3 I - 2 52 


•0947 


1.9 


•5263 


•598 


•634 


1 -137 


1.205 


36.421 


37.422 


.1104 


•1134 


34.127 


.1034 


2 .0 


.5000 


•574 


.612 


1. 149 


1.223 


39-53° 


40.733 


.1198 


•1235 


36.855 


.1117 


2. 1 


.4762 


•552 


•59o 


1. 160 


1.240 


42.536 


43.897 


.1289 


•I330 


39.450 


.1196 


2.2 


•4545 


•532 


•57i 


1. 171 


1.259 


45-407 


46.988. 


.1376 


. 1424 


41.912 


.1270 


2 -3 


.4348 


•5i4 


•553 


1. 181 


1.273 


48.199 


49.970 


.1461 


•1514 


44.287 


.1342 


2.4 


.4166 


.496 


•537 


1. 191 


1.289 


50.884 


52.878 


•1542 


.1602 


46.548 


.1411 


2 -5 


.4000 


.480 


.522 


1 .202 


1.304 


53-462 


55.676 


. 1620 


.1687 


48.720 


.1476 


2.6 


.3846 


.466 


.508 


I.2II 


I-3I9 


55-988 


58.402 


.1697 


.1769 


50.805 


•1539 


2.7 


•3704 


.452 


•493 


I.220 


1-334 


58.434 


61.054 


.1771 


.1850 


52.811 


. 1600 


2.8 


•357i 


•439 


.481 


I.229 


1.348 


60 . 800 


63-651 


.1843 


.1929 


54.745 


.1659 


2.9 


.3448 


• 427 


.469 


I.237 


1.362 


63.086 


66.175 


.1912 


.2006 


56.612 


•1715 


3-° 


•3333 


.415 


.458 


I.246 


i-375 


65-3I9 


68.626 


.1979 


.2080 


58.414 


.1770 


3-i 


.3226 


•405 


.448 


I.254 


1.388 


67.499 


71.158 


• 2045 


.2156 


60.157 


.1823 


3-2 


•3125 


•394 


.438 


I.262 


1. 401 


69 . 62 6 


73.400 


.2110 


.2224 


61.845 


.1874 


3-3 


• 3°3° 


.385 


.428 


I.270 


1. 414 


71.700 


75-686 


•2173 


.2294 


63.481 


.1924 


3-4 


.2941 


•376 


.419 


I.277 


1.426 


73.72o 


77.936 


•2234 


.2362 


65.087 


.1972 


3-5 


.2857 


•367 


.411 


I.285 


1.438 


75-688 


80.131 


.2294 


.2428 


66.610 


.2019 


3-6 


.2778 


•359 


.403 


I.292 


1.450 


77.628 


82.307 


•2352 


•2494 


68.108 


.2064 


3-7 


.2703 


•35i 


•395 


I.299 


1. 461 


79.5i6 


84.411 


.2410 


•2557 


69.564 


.2108 


3-8 


.2632 


•343 


.388 


I.306 


1-473 


81.350 


86.496 


.2465 


.2621 


70.982 


.2151 


3-9 


.2564 


•337 


•381 


I-3I3 


1.484 


83.158 


88.544 


.2520 


.2683 


72.364 


.2193 


4.0 


.2500 


■33° 


•374 


I-3I9 


1-495 


84.939 


90.510 


•2574 


•2 743 


73.710 


•2234 


4.1 


• 2439 


•323 


•3 6 7 


I.326 


1.506 


86.694 


92.472 


.2627 


.2802 


75-023 


.2274 


4.2 


.2381 


•3i7 


.361 


i-33 2 


1. 516 


88.395 


94-434 


.2678 


.2862 


76.304 


•2312 


4-3 


.2326 


•3ii 


•355 


1-339 


1.526 


90.043 


96.346 


.2729 


.2919 


77.555 


•235o 


4-4 


.2273 


.306 


•349 


1-345 


1-537 


91.691 


98.202 


.2779 


.2976 


78.776 


.2387 


4-5 


.2222 


.300 


•344 


i-35i 


i-547 


93-312 


100.012 


.2828 


•303I 


79.972 


.2424 


4-6 


.2174 


•295 


•338 


1-357 


i-557 


94.882 


101.823 


.2875 


.3085 


81. 141 


-2459 


4-7 


.2128 


.290 


■333 


*-3 6 3 


1.566 


96.424 


103.616 


.2922 


.3140 


82.284 


-2494 


4.8 


.2083 


.285 


.328 


1.368 


I.576 


97.966 


io5-37i 


.2969 


•3193 


83.404 


.2528 



73 



TABLE I (Continued). 



I 


2 


3 


4 

•3 2 4 
•3i9 
•3i5 


5 


6 


7 [ 8 


1 9 


10 


11 J 12 


4.9 

5-i 


.2041 
.2000 
.1961 


.280 
.276 
.272 


i-374 

1.380 

1.385 


1.58): 

1-595 
1.604 


99.481 
100.943 
102.405 


107. 109 
108. 811 

110.493 


•3015 
•3059 
•3103 


.3246 

•3297 
•3348 


84.500 

85-574 
86.627 


•2561 

• 2 593 
•2625 


5-2 
5-3 
5-4 


.1923 
.1887 
.1852 


.267 
.263 

•259 


.310 
.306 
.302 


i-39i 
1.396 
1. 401 


1-613 
1 .622 
1-631 


103.841 
105.260 
106.673 


112. 157 
113.830 
115.440 


•3147 
.3180 

•3 2 3 2 


•3398 
•3449 
•3498 


87.660 
88.673 
89.666 


•2657 
.2687 
.2717 


5-5 
5-6 
5-7 


.1818 
.1786 
•1754 


.256 

.252 
.248 


.298 
.294 
.291 


1.406 
1. 411 
1. 416 


1.640 
1.648 

i-657 


108.013 

i°9-353 
110.683 


117. 010 
118.570 
120. 114 


•3273 
•33i4 
•3354 


•3546 

•3593 
.3640 


90 . 642 
91.600 
92.541 


•2747 
.2776 
.2805 


5-8 

5-9 
6.0 


.1722 
.1695 

.1677 


•245 
.242 

.238 


.287 
.284 
.280 


1. 42 1 
1.426 
i-43i 


1.665 
1.673 
1. 681 


112.003 

ii3-3°5 
114-581 


121.632 
123.150 
124.640 


•3394 
•3433 
•3472 


.3686 
•3732 

•3777 


93.466 

94-375 
95-271 


•2833 
.2860 
.2887 


6.1 
6.2 

6-3 


.1639 
.1613 

•1587 


• 2 35 
.232 

.229 


•277 
.274 

.271 


1.436 
1.440 

1-445 


1.689 
1.697 
i-7o5 


115-831 
117.080 
118.303 


126. 113 
127.576 
129.030 


•35io 
•3548 

.3585 


.3822 
.3866 
.3910 


96.147 
97.012 
97.863 


.2914 

.2940 
.2966 


6.4 

6-5 
6.6 


.1562 
•1538 


.226 
.223 
.221 


.268 
•265 
.262 


1.449 
1-454 
1.458 


*-7i3 
1. 721 
1.728 


IJ 9-573 
120.723 

121.920 


130.466 
131.880 
133-30° 


.3622 
•3658 
.3694 


•3953 
•3997 
•4039 


98.700 

99.524 

100.336 


.2991 
.3016 
.3040 


6.7 
6.8 
6.9 


.1492 
.1471 
.1449 


.219 
.216 
.213 


•259 
.256 

•254 


1.464 
1.467 
1. 471 


1.736 
1.744 
i-75i 


123.063 
124.205 
125.348 


134.710 
136.090 
i37-45o 


•3729 
•3764 
•3799 


.4082 
.4124 
.4165 


101.134 
101.920 
102 . 700 


•3065 
.3088 
.3112 


7.0 

7-i 

7.2 


.1428 
.1408 
.1389 


.211 
.208 
.206 


.251 
.249 
.246 


1.476 
1.480 
1.484 


i-758 
1.766 

i-773 


126.492 
127.608 
128.708 


138.800 
140. 120 
141.430 


•3833 
•3867 
•39oo 


.4206 
.4246 
.4286 


103.465 
104.219 
104.963 


•3135 
•3158 
.3181 


7-3 

7-4 

7-5 


.1370 

•i35i 
-.1333 


.204 
.202 
.199 


.244 
.241 

•239 


1.488 
1.492 
1.496 


1.780 
1.787 
1.794 


129.789 
130.878 
131. 941 


142.710 
143-979 
145-239 


•3933 

.3966 

•3998 


•4327 
•4363 
.4401 


105.696 
106.420 
107.133 


.3203 

•3225 
•3246 


7.6 

7-7 
7.8 


.1316 
.1299 
. 1282 


.197 
•i95 
•i93 


•237 
•235 
■ 2 33 


1.500 
1.504 

1.508 


1. 801 
1.807 
1. 814 


132.995 
134.043 
135-063 


146.489 

147.732 
148.976 


.4030 
.4062 
•4093 


•4439 
•4477 
•4514 


107.837 
108.539 
109.219 


.3268 
•3289 
•33io 


7-9 
8.0 
8.1 


.1266 
.1250 
.1236 


.191 
.189 
.188 


.231 
.228 
.226 


1. 512 
1. 516 

I-5I9 


1. 821 
1.828 
1.834 


136.091 
137. no 
138. in 


150.217 
151.427 
152-633 


.4124 

•4i55 
.4185 


•4552 

.4589 
.4625 


109.896 
110.565 
in. 225 


•333° 
•335° 
•337° 


8.2 

8-3 
8.4 


.1220 
.1205 
. 1 190 


.186 
.184 
.182 


.224 
.223 
.221 


i-5 2 3 
i-5 2 7 
i-53i 


1. 841 

1.847 
1.854 


I39-093 
140.076 

141.060 


153-823 
155.010 
156.178 


•4215 
.4245 
•4275 


.4661 
.4698 
•4733 


in. 875 
112.522 
113 158 


•339o 
.3410 

.3429 


8-5 
8.6 

8-7 


.1176 
.1163 
.1149 


.180 
.179 
.177 


.219 
.217 
.215 


i-534 
1-538 
1 -54i 


1. 861 
1.867 

1-873 


142.017 
142.974 

143-931 


157.348 
158.508 
159.658 


• 4304 

•4333 
.4362 


.4768 
.4804 
.4838 


113.788 
114. 410 
115.023 


.3448 
•3465 
•3487 


8.8 
8. 9 
9.0 


.1136 
.1124 
. mi 


.176 
.174 
.172 


.214 
.212 
.210 


i-545 
1.548 

i-552 


1.879 
1.885 
1. 891 


144.862 
145.780 
146.700 


1 60 . 800 
161.927 

163.041 


.439o 
.4418 
.4446 


•4873 

.4906 

.4941 


115-633 
116.233 
116.827 


•3504 
•3522 
•354o 


9.1 

9.2 

9-3 


.1099 
.1087 
.1072 


.171 
. 170 
.168 


.208 
.207 
.205 


i-555 
i-559 
1.562 


1.897 
1.903 
1.909 


147.627 

148.557 
149-554 


164. 147 
165.236 
166.334 


•4474 
.4502 

•4532 


• 4974 

•5oo7 
.5041 


ii7-4i5 
117.996 
118. 571 


•3558 
•3576 
•3593 


9.4 

9-5 
9.6 


. 1064 
.1058 
. 1042 


.167 

.165 
.164 


.204 
.202 
.201 


1-565 
1.569 

i-572 


i-9i5 
1. 921 

1.927 


150.312 
151. 188 
152.066 


167.431 
168.520 
169.589 


•4555 
.4582 

.4609 


•5074 
•5107 

•5139 


119-138 
119.702 
120.259 


.3610 
.3627 
• 3644 


9-7 
9.8 

9-9 


.1031 
. 1020 
. 1010 


.162 
.161 
. 160 


•299 
.298 
.296 


i-575 I -933 
i-578i.939 
1. 5821. 944 


152.944 
153-794 
154-645 


170.650 
171.700 

172.754 


•4635 
.4661 

.4686 


•5i7i 
•5213 

•5235 


120.810 

121-355 
121.895 


.3661 

•3677 
•3693 


10. 


.1000 


•i59 


•295 


1. 5851. 950 


155-495 


173.789 


.4712 


.5266 


122.429 


.3710 



74 



NOTES ON TABLE II. 

The purpose of this table is to determine the weight of air compressed 
by a machine of known cubic feet capacity. It is to be used in connection 
with Table I for determining power or work. 

The barometric readings and elevations are made out for a uniform 
temperature of 6o° F. and are subject to slight errors but not enough to 
materially affect results. Table V gives more accurately the relation be- 
tween elevation temperature and pressure. 



TABLE II. — WEIGHTS OF FREE AIR UNDER VARIOUS 

CONDITIONS 



Approximate Baro- 
metric Reading. 
T=6o. 


0) 

\-> 

a> 

°C 


a 
in 


S 

< 


Weight of One Cubic Foot at Given 
Temperature (Fahr.) 


Approximate Eleva- 
tion. T=6o°. 


-20° 


oo° 


20° 


40° 


6o° 


8o° 


IOO° 


30-52 
30-32 
30.12 


15.0 
14.9 
14.8 


.09211 
.09150 
. 09089 


.08811 
•08753 

.08694 


.08444 
.08388 
•O833I 


.08l08 
.08054 
. 08000 


.07796 

.07744 
•07693 


.07508 
.07458 
.07408 


.07240 
.07192 

•07144 


— 600 

— 400 

— 200 


29.91 
29.71 
29.50 


14.7 
14.6 

14-5 


.09027 
.08965 
•08903 


.08635 

.08576 
•08517 


•08275 
.082I9 
.08l63 


•07945 
.O7895 

•07837 


.07640 
.07589 

•0753° 


•07358 

.07308 
•07258 


.07095 
•07047 

. 06999 


00 
200 
400 


29.30 
29.10 
28.90 


14.4 

14-3 
14.2 


.08842 
.08781 
.08719 


.08458 

.08400 

.08341 


.08l07 
.0805O 
•07994 


.O7783 
.O7729 
.O7675 


.07484 
.07432 
•07380 


.07208 
.07158 
.07108 


.06950 

.06902 

.06854 


600 

800 

1000 


28.69 

28.49 
28.28 


14. 1 
14.0 

13-9 


.08659 
.08597 
•08535 


.08282 

.08224 

.08165 


.O7938 
.O7882 
•07825 


.O762I 
.O7567 
•07513 


.07329 

.07277 
•07225 


•07058 
. 06008 
.06957 


.06806 

.06758 

.06709 


I200 
1400 
1600 


28.08 
27.88 
27.67 


13.8 

i3-7 

13.6 


.08474 
.08412 
.08351 


.08106 

.08048 

.07989 


.O7769 
.07713 
-O7656 


•07459 
•07405 

•07350 


•07173 
.07120 
.07068 


.06907 
.06857 
.06807 


.06661 
.06612 

.06564 


1800 
2000 
2100 


27.47 
27.27 
27.06 


i3-5 
13-4 
13-3 


.08289 
.08228 
.08167 


.07930 
.07871 

.07813 


.O760O 

•07544 
.O7487 


.O7296 
.07242 
.O7189 


.07016 
.06965 
•06913 


.06757 
.06707 
•06657 


.06516 

.06468 

.06420 


2300 
2500 
2700 


26.86 
26.66 

26.45 


13.2 

i3-i 
13.0 


.08106 
. 08044 
•07983 


•07754 
.07695 
.07637 


•07431 
•07375 
.07319 


•07135 
.O7080 

.07026 


.06861 
. 06809 
.06757 


.06607 
•06557 
•06507 


.06371 
•06323 

.06274 


2900 
3100 
3300 


26.25 
26.05 

25.84 


12.9 
12.8 
12.7 


.07921 
.07860 
.07798 


•07578 
.07518 

.07460 


.07262 
.07206 
•07150 


.06972 
.06918 
.06862 


•06705 
.06652 
. 06600 


•06457 
.06407 
•06357 


.06226 
.06178 
.06130 


3500 

3700 
4000 


25.64 

25-44 
25-23 


12.6 
12.5 
12.4 


•07737 
.07676 
•07615 


.07401 

•07343 

.07284 


.07094 
•07038 
.06981 


.068l0 
•O6756 
.06702 


.06549 
.06497 
•06445 


.06307 
•06257 
.06207 


. 06082 

•06033 
•05985 


4200 
4400 
4600 



75 



76 



COMPRESSED AIR 



TABLE II. — Continued. 



25-03 
24.83 

24.62 


12.3 
12.2 
12. 1 


•07553 
.07492 

.07430 


.07225 
.07166 
.07108 


.06925 
. 06868 
.06812 


. 06648 
.06594 
.06540 


• 06393 
•06341 
.06289 


.06157 
.06107 
.06057 


•05937 
.05889 

. 05840 


4800 
5000 
5200 


24.42 

24.22 
24.01 


12.0 
11. 9 
11. 8 


.07369 

•07307 
.07246 


.07049 
. 06990 
.06932 


•06756 
. 06699 
.06643 


. 06486 
•06432 
.06378 


.06237 
.06185 
•06133 


. 06007 

•05957 
.05907 


•05792 
•05744 
.05696 


54oo 
5600 
5800 


23.81 

23.60 

23.40 


11. 7 
11. 6 

n-5 


.07184 
.07123 
.07061 


.06873 
.06812 
•06755 


.06587 
•06530 
.06474 


.06324 
.06270 
.06216 


.06081 
.06029 

•05977 


•05857 
■ 05807 

•05757 


•05647 
•05599 
•05551 


6100 
6300 
6500 


23.20 
22.99 
22.79 


11. 4 

"■3 

11. 2 


.07000 
.06938 
.06877 


.06693 
.06638 
..06579 


.06418 
.06362 
.06305 


.06161 
.06108 
. 06054 


•05925 

•05873 
.05821 


•05707 
•05656 

.05606 


.05502 

•05454 
.05406 


6800 
7100 
7300 


22.59 
22.38 

22. l8 


11. 1 
11. 
10.9 


.06816 

•06754 
.06692 


.06520 
. 06462 
.06403 


.06249 
•06193 
.06136 


. 06000 

.05945 
.05891 


.05769 

•05717 
.05665 


•05556 
•05506 
•05456 


•05358 
•05310 
.05261 


7600 
7900 
8100 


21.98 
21.77 
21-57 


10.8 
10.7 
10.6 


.06632 
•06571 
.06510 


.06344 
.06285 
.06226 


. 06080 
.06024 
.05968 


•05837 
•05783 
.05729 


•05613 

•05561 
•05509 


.05406 

•05356 
.05306 


•05213 
.05164 
.05116 


8400 
8600 
8900 


21-37 
21 . l6 
20.96 


10.5 
10.4 
10.3 


. 06448 
.06386 
.06325 


.06168 
.06109 
.06050 


.05911 
•05855 
•05799 


•05675 
.05621 

.05567 


•05457 
•05405 
•05353 


•05256 
.05206 
.05156 


.05068 
.05020 
.04972 


9100 
9400 
9600 


20. 76 
20.55 
20.35 

■ 


10.2 

10. 1 
10. 


.06263 
.06202 
.06141 


.05991 

•05933 
•05874 


•o5743 
.05686 

.05630 


. 55i3 

•05459 
•05405 


•05301 
.05249 
.05198 


.05106 
•05056 
.05006 


.04923 

.04875 
.04827 


9900 

IOIOO 

10400 


20. 15 
19.94 
19.74 


9.9 
9.8 

9-7 


.06079 
.06017 

•o595 6 


.05816 

•o5757 
.05698 


•05572 

•05517 
.05461 


•o535i 
•05297 
.05243 


.05146 

•05094 
•05041 


•04956 
. 04906 
.04856 


•04779 

•04730 
.04682 


10700 
I IOOO 
1 1 200 


19-53 
19-33 
I9-I3 


9.6 

9-5 
9.4 


•05894 

•°5 8 33 
.05772 


•05639 

.05580 
.05522 


•05404 
.05348 
.05292 


.05188 

•05134 
.05081 


.04990 

•04937 
.04886 


. 04806 
.04756 
.04706 


•04633 
•04585 
.04538 


1 1500 

1 1 800 

1 2 IOO 


18.93 

18.72 
18.52 


9-3 
9.2 

9.1 


.05711 

•05649 

•05587 


.05463 

•05404 
•05345 


.06236 

•05179 

•05123 


.05027 

.04972 
.04918 


•04834 
.04782 
.04730 


■04655 
. 04605 

.04555 


. 04489 
. 04440 
.04392 


I24OO 
1270O 
I3OOO 


18.31 


9.0 


•05526 


.05286 


.05067 


.04864 


.04678 


■04505 


• 04344 


13400 



NOTE ON TABLE III. 
The table is designed to compute readily weights of compressed air by 



formula 12, Art. 8, viz., w = 
inch the formula becomes w 



53-17' 
= 144 X p 

53.I7X/' 



If p is given in pounds per square 



The value — can most readily be obtained with the slide rule. 



TABLE III. — WEIGHTS OF COMPRESSED AIR 

Pounds per Cubic Foot. 

P 
The Ratio - is for absolute pressure in pounds per square inch and abso- 
z> 

lute temperature Fahrenheit. (See Note at foot of previous page.) 



t 

t 


w 


P 
t 


w 


t 
t 


w 


t 

t 


w 


.000 

.005 

.010 


. 0000 

■013s 

.0271 


2.55 
260 
265 


.6906 
.7041 

.7177 


5io 

515 

520 


1-3813 
1-3947 
1.4083 


•765 

•77o 
•775 


2.0718 
2.0853 
2 . 0988 


.015 

.020 

.025 


.0406 

.0542 
.0677 


270 

275 
280 


•73i2 
•7447 
.7583 


5 2 5 
53° 
535 


1 .4219 

1-4355 
1 . 4490 


.780 

•785 
•79o 


2.1125 
2 . 1260 
2.1395 


.030 

•035 

.040 


.0813 

.0948 

.1083 


285 
290 

295 


.7719 
.7852 
•7989 


54o 

545 
55o 


1.4625 
1.4760 
1.4895 


•795 
.800 
.805 


2-153° 
2.1665 
2. 1798 


• 045 
.050 

•°55 


.1218 

• 1354 

.1489 


300 

305 
310 


.8125 
.8260 
•8395 


555 
560 

565 


1 • 5°3° 
1. 5166 

i-53i2 


.810 

.815 

.820 


2.1950 

2.2071 
2 . 2207 


.060 
.065 

.070 


.1625 

.1760 

.1896 


315 
320 

325 


•8531 
.8666 
.8801 


57o 
575 
580 


1-5437 
1-5572 
I-5707 


•825 
•830 
•835 


2-2343 
2 . 2480 
2 .2615 


•o75 
.080 
.085 


.2031 

.2166 
. 2*302 


33o 
335 
34o 


•8937 
.9072 
.9208 


585 
59o 
595 


1-5843 
1 . 5980 

1-6115 


.840 

•845 
•850 


2.2750 
2.2885 
2 .3020 


.090 

•095 
. 100 


.2437 
•2573 • 

.2708 


345 
35° 
355 


•9343 
.9478 

.9613 . 


600 
605 
610 


1 .6250 

1.6385 
1.6520 


•855 
.860 
.865 


2-3I55 
2 .3290 

2.3425 


.105 
. no 
•ii5 


• 2843 
.2979 
.3114 


360 

365 
37° 


•9749 

.9884 

1 . 0020 


615 
620 
625 


1 . 6654 
1 .6792 
1.6927 


.870 

•875 
.880 


2-356i 

2.3698 

2-3833 


. 120 
.125 
.130 


•3250 • 

•3385 ■ 

•3520 


375 

380 

385 


i-oi55 

1 .0290 

1.0425 


630 

635 
640 


1 . 7062 
1. 7198 

1 • 7333 


.885 
.890 
.895 


2.3970 

, 2.4105 

2 . 4240 


•135 
. 140 

•145 


•3656 

•379 2 
.3927 


39° 

395 
400 


1. 0561 
1.0697 
1-0833 


645 
650 

655 


1 . 7468 
1 . 7603 
1-7739 


.900 

•905 
.910 


2-4375 
2.4510 

2 ■ 4645 


.150 

•155 
. 160 


.4062 
.4197 
•4333 


405 
410 

4i5 


1 . 0968 
1. 1 103 

1 . 1240 


660 

665 
670 


I-7875 
1. 8010 
1. 8145 


•9i5 
.920 

•9 2 5 


2.4780 
2.4917 
2.5052 


.165 

.170 

•175 


.4468 
.4603 
•4739 


420 
425 
43° 


1 -1375 
1.1510 

1-1645 


675 
680 
685 


1.8280 
1-8415 
1-8550 


•93o 

•935 
.940 


2.5187 

2.53 2 3 
2-5459 


.180 

•185 
. 190 


•4875 
.5010 

•5145 


435 
440 

445 


1. 1780 
1. 1917 
1. 2052 


690 

695 
700 


1 . 8680 
1.8822 
1.8959 


•945 

•95° 
•955 


2-5594 
2-573° 
2.5865 


•195 
.200 

.205 


•5281 
.5416 

•555i 


•45° 

•455 

460 


1. 2177 
1.2323 
1-2457 


■705 
.710 

•7i5 


1 . 9094 
1 .9229 
1 9365 


.960 
•965 

•97o 


2 . 6000 
2-6135 
2 . 6270 


. 210 
.215 
.220 


.5687 
.5822 
•5958 


.465 
.470 

•475 


1-2594 
1 .2730 
1.2865 


.720 
•725 
•73° 


1 . 9500 

I-9635 
1.9770 


•975 
.980 

•985 


2 - 6405 
2.6541 
2 . 6670 


.225 
.230 
•235 


.6094 
.6229 
.6364 


.480 

.485 
.490 


1 . 3000 

i-3i35 
1.3270 


•735 
.740 

•745 


1.9905 
2 . 0042 
2.0177 


990 

•995 
1 000 


2 6813 
2 . 6949 
2 . 7084 


.240 

•245 
.250 


.6499 
•6635 
.6771 


•495 

.500 

• 5o5 


i-34i6 
1-3542 
1-3677 


•75o 
•755 
.760 


2.0312 
2 . 0448 
2.0582 







77 



TABLE IV. * — SPECIAL TABLE RELATING TO STAGE COM- 
PRESSION FROM FREE AIR AT 14.7 POUNDS PRESSURE 
AND 62 TEMPERATURE. 



Compression adiabatic but cooled between stages. 



tn 
w 

s 


a 

.9 
'«> 

CO 



Ih 

ft 

a 




£ ft 


Single Stage. 


Two Stage. 




> 

I B 

S 8 


P 5 • 


1 <+-{ «j 

a ° 3 

O +J a 

O fo S 
;^ 


c 


'xn 

03 

<U (0 

ft d 


g iS 


to Com- 
u. Ft. of 
r Minute. 


ft, 

bO 
a 

O 




O 

.9 

d 


eight of On 
Foot at T 
ture 6a°F 


ft S 


orse Power 
press One C 
Free Air pe 


§ w 

M 

■£ .a 

d w 


inal Temper 
Each Stag 
renheit. 


orse Power 
press One C 
Free Air pe 






* 


§ 


s 


n 


Ph 


Ph 


E 


Pg 


r 


W 


M.E.P. 


t\ 


H.P. 


Vr 


r 2 


H.P. 


5 


i-34 


.1020 


4-5o 


108 


.0197 








10 


1.68 


.1279 


8.30 


144 


.0362 








IS 


2.02 


•1537 


11. 51 


177 


.0045 








20 


2.36 


.1796 


14.40 


207 


.0628 








2 5 


2. 70 


•2055 


17.00 


235 


.0742 








3° 


3-°4 


•2313 


19.40 


259 


.0845 








35 


3-3* 


.2572 


21.65 


280 


.0944 








40 


3-7 2 


.2831 


23.60 


303 


.1030 








45 


4.06 


.3090 


25-50 


321 


. 1112 








5o 


4.40 


.3348 


27.50 


341 


•1195 


2.10 


180 


.1063 


55 


4-74 


.3607 


29. 10 


358 


.1268 


2.17 


189 


.1123 


60 


5.08 


.3866 


30-75 


373 


•1339 


2.25 


196 


.1184 


65 


5-42 


.4124 


32.30 


392 


.1408 


2-33 


200 


•1235 


70 


5-76 


.4383 


33- 80 


405 


.1472 


2.40 


207 


.1286 


75 


6. 10 


.4642 


35-i8 


420 


•1532 


2.47 


214 


.1329 


80 


6.44 


.4901 


3 6 -55 


434 


.1590 


2-54 


222 


•1372 


85 


6.78 


•5159 


37-9o 


447 


.1650 


2.60 


227 


.1410 


90 


7.12 


.5418 


39.10 


461 


•1705 


2.67 


233 


.1462 


95 


7.46 


.5676 


40.35 


473 


.1758 


2-73 


238 


.1500 


100 


7.80 


•5935 


41.65 


485 


.1812 


2-79 


242 


.1542 


105 


8.14 


.6194 


42.30 


497 


.1841 


2.85 


246 


.1578 


no 


8.48 


•6453 


43-75 


508 


.1908 


2.90 


251 


.1615 


IJ 5 


8.82 


.6712 


45.16 


519 


.1965 


2.99 


256 


.1648 


120 


9.16 


.6971 


46.00 


530 


. 2008 


3.02 


259 


.1681 


125 


9-5o 


.7230 


47-o5 


540 


.2045 


3.08 


262 


.1710 


130 


9.84 


.7488 


47.80 


550 


.2085 


3-14 


266 


.1740 


135 


10.18 


•7747 


48.85 


560 


•2135 


3.19 


269 


•1775 


140 


10.52 


.8005 


49.90 


569 


.2176 


3-24 


272 


.1810 


145 


10.86 


.8264 


51.00 


578 


.2220 


3- 2 9 


276 


•1837 


150 


11.20 


.8522 


5i-7o 


587 


•2255 


3-35 


280 


.1865 



* The table is limited to the special initial condition of air specified in 
the caption. The assumption of 14.7 as atmospheric pressure makes the 
weights and work a little in excess of average conditions. However, it is a 
valuable and very instructive table. 

78 



PLATES AND TABLES 



79 



TABLE IV {Continued). 



<6 

u 
3 
in 
in 

<L> 
U 

a. 


d 


Cfl 

in 
<u 
i- 

0, 

s 








fe 
x> 

O "O 

°^ 



r; q 


Two Stage. 


Three Stage. 


& 



in 

3 « 

u so 



■8 a 


eg 

£: so 
a $ 


1 <+-< 4) 

fe 3 

> C v- 
O O < 

« " 8 


c 


to 

£ <u 

■~ bo 
a ai 

£ w 




pq 


•3 J 

« of 
4) •- 


1 u-i 4) 

S ° 3 
fe 3 

d s 

a> <u a 
fe C «- 
O O < 
A* in „, 
« M S 


0) 

as 




a) 


M O 


3 a 

cij 




4) 4) <U 
W l_ I- 







4) 4) OJ 


O 


rt 


P4 


£ 


w 


ti 


£ 


w 


Pg 


r 


ze 


(r)» 


r 2 


H.P. 


(r)» 


r 3 


H.P. 


IOO 


7-8 


•593 6 


2.79 


242 


•1542 


1.98 


176 


.1450 


150 


11. 2 


.8522 


3 


35 


280 


.1865 


2 


.24 


200 


• 1752 


200 


14.6 


I. IIIO 


3 


82 


308 


.2110 


2 


■44 


215 


.1965 


250 


18.0 


1.3697 


4 


24 


332 


•2315 


2 


62 


226 


.2140 


300 


21.4 


1 . 6285 


4 


63 


353 


.2490 


2 


78 


241 


.2295 


35° 


24.8 


1.8872 


4 


98 


37° 


.2640 


2 


92 


251 


.2418 


400 


28.2 


2.1459 


5 


3i 


386 


.2770 


3 


04 


259 


• 2535 


45° 


31.6 


2 . 4048 


5 


61 


399 


.2895 


3 


16 


267 


.2630 


500 


35-o 


2 . 6634 


5 


9 1 


412 


.2915 


3 


27 


275 


.2730 


5So 


38.4 


2.9221 








3 


37 


281 


.2830 


600 


41.8 


3.1810 








3 


47 


287 


.2910 


650 


45- 2 


3-4395 








3 


56 


292 


.2960 


700 


48.6 


3 . 6982 








3 


64 


297 


• 3025 


75o 


52.0 


3-957° 








3 


73 


302 


.3090 


800 


55-4 


4-2155 








3 


80 


307 


• 3150 


850 


58.8 


4-4745 








3 


83 


312 


.3210 


900 


62.2 


4-733° 








3 


96 


316 


.3260 


95° 


65.6 


4.9920 








4 


°3 


320 


•3315 


1000 


69.0 


5-2510 








4 


10 


324 


•3360 


1050 


72.4 


5-5095 








4 


17 


328 


.3400 


1 100 


75-8 


5-7684 








4 


23 


331 


•3445 


1 150 


79.2 


6.0270 








4 


29 


334 


• 3490 


1200 


82.6 


6.2855 








4 


36 


337 


• 3525 


1250 


86.0 


6-5445 








4 


41 


341 


• 3570 


1300 


89.4 


6.8030 








4 


47 


344 


• 3615 


i35° 


92.8 


7.0620 








4 


52 


347 


.3660 


1400 


96.2 


7.3210 








4 


58 


350 


•3685 


145° 


99.6 


7-5795 








4 


64 


353 


• 3710 


1500 


103.0 


7.8382 








4 


70 


356 


• 3740 


i55o 


106.4 


8.0965 








4- 


75 


359 


.3780 


1600 


109.8 


8-355o 








4 


79 


361 


.3820 


1650 


113. 2 


8.6140 








4 


83 


363 


•3850 


1700 


116. 6 


8.8730 








4 


87 


365 


.3880 


i75o 


120.0 


9.1320 








4 


93 


367 


• 3915 


1800 


123.4 


9.3900 








4 


97 


369 


• 3940 


1850 


126.8 


9.6485 








5 


02 


371 


• 3965 



TABLE V. — VARYING PRESSURES WITH ELEVATIONS. 

h 
Solution of formula 17, Art. 17, viz. logi /> a = 1. 16866 — 



122.4 / 



Elevation in Feet. 


Pressure 


in Pounds per Square Inch. 










Temp. 5o°F. 


Temp. 3 5°F. 


Temp. 2o°F. 





14.70 


14.70 


14.70 


IOOO 


14.17 


14-15 


14.14 


2000 


13.66 


I3-63 


13-99 


3000 


13.16 


13.12 


13.07 


4000 


12. 69 


12.63 


12-57 


5000 


12.23 


12.16 


12.09 


5280 


12. IO 


12.03 


11 .96 


6000 


II.78 


II. 71 


11.63 


7000 


II.36 


11 .27 


II. 18 


8000 


IO.95 


10.85 


io-75 


9000 


i°-55 


10.45 


10.33 


1 0000 


10. 17 


10.06 


9.94 


12500 


9.28 


915 


9.02 


15000 


8.46 


8.32 


8.18 



TABLE VI.* — HIGHEST LIMIT TO EFFICIENCY WHEN 
COMPRESSED AIR IS USED WITHOUT EXPANSION, 
ASSUMING ATMOSPHERIC PRESSURE = 14.5 POUNDS 
PER SQUARE INCH. 



r 


h 


E 


r 


h 


E 


r 


h 


E 


1 .2 


6.66 


91.4 


5-2 


140.0 


49 .0 


9.2 


2 73-3 


40.2 


1.4 


*3-3 


84.9 


5-4 


146.6 


48 


3 


9.4 


280.0 


39 


9 


1.6 


20.0 


79.8 


5-6 


153-3 


47 


7 


9.6 


286.6 


39 


6 


1.8 


26.6 


75-6 


5-8 


160.0 


47 





9.8 


293-3 


39 


3 


2.0 


33-3 


72.0 


6.0 


166.6 


46 


5 


10. 


300.0 


39 





2.2 


40.0 


69.2 


6.2 


173-3 


46 





10.25 


3°% -3 


38 


6 


2.4 


46.6 


66.7 


6.4 


180.0 


45 


5 


10.50 


316.6 


38 


5 


2.6 


53-3 


61.9 


6.6 


186.6 


45 





i°-75 


3 2 5-° 


38 





2.8 


60.0 


62.4 


6.8 


193-3 


44 


5 


11.00 


333-3 


37 


9 


3-° 


66.6 


60.7 


7.0 


200.0 


44 





11.25 


341.6 


37 


7 


3- 2 


73-3 


59-i 


7.2 


206.6 


43 


6 


11.50 


35o-o 


37 


4 


3-4 


80.0 


57-8 


7-4 


213-3 


43 


1 


n-75 


353-3 


37 


1 


3-6 


86.6 


56.4 


7.6 


220.0 


42 


8 


12.00 


366.6 


36 


9 


3-» 


93-3 


55- 2 


7.8 


226. 6 


42 


4 


12.25 


375 -o 


36 


7 


4.0 


100 .0 


54-i 


8.0 


233- 3 


42 





12.50 


3^3-3 


36 


4 


4.2 


106. 6 


53-i 


8.2 


240.0 


4i 


7 


12.75 


391.6 


36 


2 


4.4 


H3-3 


52.1 


8.4 


246.6 


4i 


4 


13.0 


400.0 


36 





4-6 


120.0 


5i-3 


8.6 


253-3 


4i 


1 


14.0 


433-3 


35 


2 


4.8 


126.6 


5o-5 


8.8 


260.0 


40 


8 


15.0 


466.6 


34 


5 


5-o 


133-3 


49-7 


9.0 


266.6 


40 


5 


16.0 


500.0 


33 


8 



* This table reveals the limit of efficiency when air is applied without 
utilizing any of its expansive energy. 

The column headed r gives the ratio of compression, while that headed h 
gives the water head equivalent to a pressure given by the ratio r on the 
assumption that one atmosphere is a pressure of 14.5 pounds per square 
inch or a water head of t>3-3 feet, this being more nearly the average condi- 
tion than 14.7, which is so commonly taken. 

It should be understood that this efficiency cannot be reached in practice, 
— it being reduced by friction of air and machinery and by clearance in 
any form of engine. 

80 



PLATES AND TABLES 



81 



TABLE VII. — EFFICIENCY OF DIRECT HYDRAULIC AIR 

COMPRESSORS. 

2.3 logi r 



Formula 26, Art. 25, viz. E = 



r — 1 



Water Head. 


Gage Pressure. 


Absolute Pres- 
sure. 


Atmospheres 

=r 


Efficiencv, 
E 


0.0 


0.0 


14-5 


1 


1 .00 


33-3 


14-5 


29.0 


2 


.69 


66.6 


29.0 


43-5 


3 


•55 


100. 


43-5 


58.0 


4 


.46 


133-3 


58.0 


7 2 -5 


5 


.40 


166.6 


72-5 


87.0 


6 


.36 


200.0 


87.0 


101.5 


7 


•33 


233-3 


101.5 


116. 


8 


•3° 


266.0 


116. 


i3°-5 


9 


.28 


300.0 


i3°-5 


i45-° 


10 


.26 



TABLE VIII. — COEFFICIENT " c» FOR VARIOUS HEADS 
AND DIAMETERS. 



d" 


i=i" 


1=2" 


i=z" 


i=A" 


1= 5 


5 

16 
1 
2" 


0.603 
0. 602 


0.606 

0.605 


0.610 
0.608 


O.613 
0.610 


O. 616 
O.613 


I 

2 


0.601 
0. 601 
0.600 


0. 603 
0. 601 
0. 600 


0.605 
0. 602 
0.600 


O.606 
O.603 
O.600 


O.607 
O.603 
O.600 


2^ 
3 


o.599 
599 


0599 
0.598 


o-599 
o-597 


O 598 
O.596 


O.598 
O 596 


3^ 


599 


597 


0.596 


o-595 


0-594 


4 

4h 


0.598 
598 


o-597 
0.596 


595 
0.596 


• 594 
o-593 


o-593 
0.592 



Table VIII gives the experimental coefficients for orifices for determining 
the weight of air passing by formula: 



Weight (<2) = 0.6299 cd 2 Jt 



Q = Weight of air passing in pounds per second. 
c = Experimental coefficient. 
d = Diameter of orifice in inches. 
* = Difference of pressure inside and outside of orifice in inches of 

water. 
/ = Absolute temperature of air back of orifice. 



82 



COMPRESSED AIR 



TABLE IX.— FRICTION IN AIR PIPES. 





Divi 


de the number corresponding to the diameter and volume 




by the ratio of compression. The result is the loss 


in pounds per 


eet o 

Air p 
te. 


square 


inch in 1000 feet of pipe. 


























Diameter of Pipe 


in Inches. 






.Q fc S 


















3 


1 
2 


3 
"4 


1 


ii 


ii 


1 1 


2 


2\ 


3 


6 


2 7-3 


35-4 


•83 


.26 












12 


108.3 


14.26 


3-3 2 


1.05 












24 




56.64 


13.28 


4.20 


1. 71 


.78 








36 




126.4 


29.86 


9-45 


3-84 


I 


75 








48 




226.6 


53-15 


16.80 


6.83 


3 


12 


1.60 






60 






84.94 


26.26 


10. 70 


4 


87 


2.50 






72 






119. 8 


37-9° 


15.40 


7 


°3 


3.62 


1. 17 




84 






163.7 


51.46 


20.90 


9 


55 


4.91 


i-59 




96 








67.21 


27.30 


12 


48 


6.41 


2.07 




I08 








85.06 


34-55 


15 


80 


8.12 


2.62 




I20 








105.0 


42.67 


19 


50 


10.00 


3-25 




I50 








164. 1 


66.53 


3° 


47 


15.66 


5.06 


1.85 


l8o 










96.00 


43 


87 


22.54 


7.28 


2.67 


2IO 










130-7 


59-71 


30.70 


9.91 


3- 6 3 


if 


2 


2§ 


3 


3§ 


4 


4i 


5 


6 


24O 


78.00 


40.09 


12.94 


4-74 


2.13 










270 


98.70 


50.72 


16.48 


6.00 


2. 70 










3OO 


E2I.8 


62.62 


20.23 


7.41 


3-33 










33° 




75 -7 8 


2 4-57 


8-97 


4-03 










360 




90.29 


29. 12 


10.67 


4.80 










39° 




105-5 


34.20 


12-53 


5-63 










420 




122.8 


39-64 


14.52 


6-53 


2.87 








45° 






45-5° 


16.67 


7-49 


3-3° 








480 






51.88 


18.97 


8-53 


3-75 








5io 






5 8 -44 


21 .42 


9.62 


4-23 








540 






65-39 


24.01 


10.79 


4-75 








57o 






73.00 


26.75 


12.02 


5- 2 9 


2-94 






600 






80.90 


29.64 


I3-32 


5.86 


3- 2 5 






660 






97.90 


35-87 


16. 12 


7.09 


3-93 






720 






116.50 


42.68 


19.19 


8-43 


4.68 






780 








50.10 


22.50 


9.00 


5-5o 


3- 2 5 




840 








58.10 


26.11 


11.48 


6-37 


3-76 




900 








66.70 


29.98 


13.18 


7-3 2 


4.32 




960 








75-88 


34-IO 


15.00 


8.32 


4.92 




1020 








85-65 


38.50 


16.93 


9.40 


5-55 


2.23 


1080 








96.04 


43-17 


18.98 


10.53 


6. 22 


2.50 


1 140 








107.00 


48. 10 


21.15 


11.74 


6-93 


2.79 


1200 










53-29 


23-44 


13.01 


7.68 


3-°9 


1320 










64.49 


28.36 


15-74 


9.29 


3-73 


1440 










76.74 


33-75 


18.73 


11.06 


4.44 


1560 










90.05 


39.61 


21.89 


12.98 


5.22 


1680 










104.45 


45-95 


25-5o 


16.78 


6.05 



TABLE IX {Continued). 



This table represents the author's formula 20, Chap. IV., 



/ 



d 5 r 



f = Loss of pressure in pounds per square inch. 
c = An experimental coefficient. 
/ = Length of pipe in feet. 
d = Diameter of pipe in inches. 
v a = Cubic feet of free air passing per second. 
r = Ratio of compression from free air. 

83 



Feet 

i Air 
lute. 






Diameter of Pipe in Inches 


































a 


4 


4* 


5 


6 


8 


10 


12 


1800 


52.73 


28.23 


17.82 


6-95 


I.65 






1920 


60.0c 


33 -3c 


ig.6t 


7.9c 


I.87 






2040 


67.74 


37-59 


22.2c 


8.92 


2. 12 






2160 


75-94 


42.15 


24.89 


10. oc 


2-37 






2280 


84.60 


46.95 


27.65 


11. 14 


2.64 






2400 


93-74 


52.02 


30.72 


I2 -35 


2-93 


0.96 




2520 




53-38 


33-87 


13.61 


3- 2 3 


1.06 




2640 




62.96 


37-17 


14.94 


3-55 


1. 16 




2780 




68.81 


40.66 


* 6 -33 


3.88 


1.27 




2880 




74.92 


44.78 


17.78 


4.22 


i-38 




3000 




81.30 


48.00 


19.29 


4.58 


1.50 




33°o 




98.37 


58.08 


23-34 


5-54 


1. 81 




3600 






69.13 


27.78 


6-59 


2. 16 


0.87 


3900 






81.13 


32.60 


7-74 


2-53 


1.02 


4200 






94.09 


37.8i 


8-97 


2.94 


1. 18 


4500 








43-41 


10.30 


3-37 


1.36 


4800 








49-39 


11.72 


3-84 


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5100 








55-76 


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4-34 


1.74 


5400 








62.51 


14.83 


4.86 


i-95 


5700 








69.62 


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27.87 


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TABLE X. 



TABLE OF CONTENTS OF PIPES IN CUBIC 
FEET AND IN U. S. GALLON. 







For 1 Foot 


in Length. 






For 1 Foot 


in Length. 




Diam. 








Diam. 






Diam. 


in Deci- 


Cubic Feet. 


Gallons of 


Diam. 


in Deci- 


Cubic Feet. 


Gallons of 


in 
Inches. 


mals of 
a Foot. 


Also Area 

in Square 

Feet. 


23 1 Cubic 
Inches. 


in 
Inches. 


mals of 
a Foot. 


Also Area 

in Square 

Feet. 


23 1 Cubic 
Inches. 


1 

4 


.0208 


.0003 


.0026 


II . 


.9167 


.6600 


4-937 


5 
16 


.0260 


.0005 


.0040 


1 

4 


•9375 


.6903 


5-i6 3 


3 

8 


•03I3 


.0008 


.0057 


1 

2 


•9583 


.7213 


5-395 


7 
16 


•0365 


.0010 


.0078 


3 

4 


.9792 


•7530 


5-633 


1 
2 


.0417 


.0014 


.0102 


12. 


1 Foot 


•7854 


5-876 


9 
16 


.0469 


.0017 


.0129 


1 
2~ 


1.042 


•8523 


6-375 


5 

8 


.0521 


.0021 


.0159 


13- 


1.083 


.9218 


6.895 


1 1 
16 


•0573 


.0026 


.0193 


1 
2" 


1-125 


.9940 


7-435 


i 
4 


.0625 


.0031 


.0230 


14. 


1. 167 


I.069 


7-997 


13 

16 


.0677 


.0036 


.0270 


1 

2" 


1 .208 


I. 147 


8.578 


7 
8 


.0729 


.0042 


.0312 


is- 


1.250 


I.227 


9. 180 


15 
16 


.0781 


.0048 


•°359 


1 

2 


1.292 


1. 310 


9.801 


I . 


• o8 33 


•0055 


.0408 


l6. 


i-333 


1.396 


10.44 


1 
4 


. 1042 


.0085 


.0638 


1 
2 


i-375 


I.485 


11. 11 


1 

2~ 


.1250 


.0123 


.0918 


i7- 


1-417 


1-576 


11.79 


3 


.1458 


.0168 


.1250 


1 

1 


1.458 


1 .670 


12.50 


2 . 


. 1 667 


.0218 


.1632 


18. 


1.500 


1.767 


13.22 


1 
4 


•i875 


.0276 


.2066 


1 
2 


1.542 


I.867 


J 3-97 


1 
2" 


.2083 


.0341 


•2550 


19. 


1.583 


1.969 


J 4-73 


3 
4 


.2292 


.0413 


•3085 


1 

2 


1.625 


2 .074 


15-52 


3- 


.2500 


.0491 


•3 6 73 


20. 


1.666 


2. 182 


16.32 


i 

4 


.2708 


.0576 


.4310 


1 

2 


1.708 


2.292 


J 7-i5 


1 

2 


.2917 


.0668 


.4998 


21. 


i-75° 


2.405 


17.99 


2 

3 


•3i 2 5 


.0767 


•5738 


1 

2 


1.792 


2.521 


18.86 


4- 


■3333 


.0873 


.6528 


22. 


1-833 


2. 640 


19-75 


i 

4 


•3542 


.0985 


•737° 


1 

2 


1-875 


2.761 


20.65 


1 
7 


•375o 


.1105 


.8263 


2 3- 


i-9i7 


2.885 


22.58 


3 

4 


•395 8 


.1231 


.9205 


1 

2 


1-958 


3.012 


2i-53 


5- 


.4167 


.1364 


1 .020 


24. 


2.000 


3.I42 


23-5° 


i 

4 


•4375 


•I503 


1 . 124 


25- 


2.083 


3-409 


25-50 


1 
5 


•4583 


.1650 


1.234 


26. 


2.166 


3.687 


27-58 


3 

4 


.4792 


.1803 


1-349 


27. 


2.250 


3-976 


29.74 


6. 


.5000 


.1963 


1.469 


28. 


2-333 


4.276 


31-99 


i 

4 


.5208 


.2130 


1-594 


29. 


2.416 


4-587 


34.3i 


1 

2 


•54i7 


• 2 305 


1.724 


30. 


2.500 


4.909 


36.72 


I 
4 


•5625 


.2485 . 


1-859 


3i- 


2-583 


5-241 


39.21 


7 


•5833 


.2673 


1.999 


3 2 - 


2.666 


5-585 


41.78 


i 

4 


.6042 


.2868 


2.144 


33- 


2.750 


5 -94o 


44-43 


£ 


.6250 


.3068 


2.295 


34 


2-833 


6-3°5 


47-17 


3 
4 


.6458 


•3275 


2.450 


35- 


2.916 


6.681 


49.98 


8. 


.6667 


-349° 


2. 611 


36. 


3.000 


7.069 


52.88 


i 

4 


•6875 


•3713 


2.777 


37- 


3-083 


7.468 


55-86 


1 
1, 


.7083 


•394o 


2.948 


38. 


3.166 


7.876 


58.92 


3 
4 


.7292 


•4175 


3-125 


39- 


3-250 


8.296 


62.06 


9- 


.7500 


.4418 


3-3°5 


40. 


3-333 


8.728 


65.29 


1 

4 


.7708 


.4668 


3-492 


41. 


3.416 


9.168 


68.58 


1 
2 


.7917 


•4923 


3.682 


42. 


3-5oo 


9.620 


71.96 


3 

4 


.8125 


•5i85 


3-879 


43- 


3-583 


10.084 


75-43 


IO. 


•8333 


•5455 


4.081 


44. 


3.666 


10.560 


79.00 


1 
4 


.8542 


•573° 


4.286 


45- 


3-75o 


11.044 


82.62 


1 

2" 


.8750 


.6013 


4.498 


46. 


3-833 


11.540 


86.32 


3 

4 


.8958 


•6303 


4.714 


47- 


3.916 


12.048 


90. 12 










48. 


4.000 


12.566 


Q4.02 



89 



TABLE XI — CYLINDRICAL VESSELS, TANKS, CISTERNS, 

ETC. 

Diameter in Feet and Inches, Area in Square Feet, and U. S. Gallons 

Capacity for One Foot in Depth. 

11 i • • i 1 cubic foot rn . -. 

i gallon = 231 cubic inches = = 0.13300 cubic feet. 

* 6 7-4805 6d 



Diam. 


Area. 


Gals. 


Diam. 


Area. 


Gals. 


Diam. 


Area. 


Gals. 


Ft. In. 


Sq. Ft. 


1 Ft. 
Depth. 


Ft. 


In. 


Sq. Ft. 


1 Ft. 
Depth. 


Ft. 


In. 


Sq. Ft. 


1 Ft. 
Depth. 


I 


•785 


5-89 


5 


5 


23.04 


172.38 


17 


6 


240.53 


1799-3 


I I 


.922 


6.89 


5 


6 


23.76 


177.72 


17 


9 


247-45 


1851.1 


I 2 


I.069 


8.00 


5 


7 


24.48 


I83.I5 


18 




254-47 


1903.6 


1 3 


I.227 


9.18 


5 


8 


25.22 


188.66 


18 


3 


261.59 


1956.8 


1 4 


I.396 


10.44 


5 


9 


25-97 


194.25 


18 


6 


268.80 


2010.8 


1 5 


1-576 


n-79 


5 


10 


26.73 


199.92 


18 


9 


276. 12 


2065.5 


1 6 


I.767 


13.22 


5 


11 • 


27.49 


205.67 


19 




283.53 


2120.9 


1 7 


I.969 


14-73 


6 




28.27 


211. 51 


19 


3 


291.04 


2177. ! 


1 8 


2. 182 


16.32 


6 


3 


30.68 


229.50 


19 


6 


298.65 


2234.O 


1 9 


2.405 


17.99 


6 


6 


33-18 


248.23 


19 


9 


306.35 


2291.7 


1 10 


2.640 


19-75 


6 


9 


35-78 


267.69 


20 




314.16 


2 350.I 


1 11 


2.885 


21.58 


7 




38.48 


287.88 


20 


3 


322.06 


2409. 2 


2 


3.142 


2 3-5° 


7 


3 


41.28 


308.81 


20 


6 


33O.06 


2469 . I 


2 1 


3-409 


2 5-5o 


7 


6 


44- 18 


33 .48 


20 


9 


338.16 


2529.6 


2 2 


3.687 


27.58 


7 


9 


47.17 


352.88 


21 




346.36 


2591.O 


2 3 


3-976 


29.74 


8 




50.27 


376.01 


21 


3 


354-66 


2653.O 


2 4 


4.276 


3i.99 


8 


3 


53.46 


399.88 


21 


6 


363.05 


2715.8 


2 5 


4-587 


34.3i 


8 


6 


56.75 


424.48 


21 


9 


371-54 


2779.3 


2 6 


4.909 


36.72 


8 


9 


60.13 


449 . 82 


22 




380.I3 


2843 . 6 


2 7 


5.241 


39- 2 i 


9 




63.62 


475.89 


22 


3 


388.82 


2908.6 


2 8 


5.585 


41.78 


9 


3 


67.20 


502.70 


22 


6 


397-61 


2974-3 


2 9 


5 -94o 


44-43 


9 


6 


70.88 


530.24 


22 


9 


406.49 


3040 . 8 


2 10 


6 -3°5 


47.16 


9 


9 


74.66 


558.51 


2 3 




4I5.48 


3108.0 


2 11 


6.681 


49.98 


10 




78.54 


587-52 


23 


3 


424.56 


3175.9 


3 


7.069 


52.88 


■10 


3 


82.52 


617.26 


2 3 


6 


433-74 


3244.6 


3 1 


7.467 


55-86 


10 


6 


86.59 


647 • 74 


23 


9 


443-OI 


33i4.o 


3 2 


7.876 


58.92 


10 


9 


90. 76 


678.95 


24 




452.39 


3384-1 


3 3 


8.296 


62.06 


11 




95- °3 


710.90 


24 


3 


461.86 


3455 -o 


3 4 


8.727 


65.28 


11 


3 


99.40 


743.58 


24 


6 


471.44 


3526.6 


3 5 


9.168 


68.58 


11 


6 


103.87 


776.99 


24 


9 


481. 11 


3598.9 


3 6 


9.621 


71.97 


11 


9 


108.43 


811 . 14 


25 




490.87 


3672.0 


3 7 


10.085 


75-44 


12 




113.1° 


846.03 


25 


3 


500.74 


3745-8 


3 8 


IO-559 


78.99 


12 


3 


117.86 


881 . 65 


25 


6 


510.71 


3820,3 


3 9 


11.045 


82.62 


12 


6 


122. 72 


918.00 


25 


9 


520.77 


3895-6 


3 10 


11. 541 


86.33 


12 


9 


127.68 


955.09 


26 




530-93 


3971.6 


3 11 


12.048 


90.13 


13 




132.73 


992.91 


26 


3 


541-19 


4048.4 


4 


12.566 


94.00 


13 


3 


137-89 


1031-5 


26 


6 


55L55 


4125.9 


4 1 


i3-o95 


97.96 


13 


6 


143.14 


1070.8 


26 


9 


562 .00 


4204. 1 


4 2 


13-635 


102.00 


13 


9 


148.49 


1110.8 


27 




572.56 


4283.0 


4 3 


14.186 


106.12 


14 




153-94 


1151-5 


27 


3 


583-21 


4362 . 7 


4 4 


14.748 


110.32 


14 


3 


159.48 


1193-0 


27 


6 


593.96 


4443 • 1 


4 5 


i5-3 2 i 


114. 61 


14 


6 


165-13 


1235.3 


27 


9 


604.81 


4524.3 


4 6 


15.90 


118.97 


14 


9 


170.87 


1278.2 


28 




6i5.75 


4606.2 


4 7 


16.50 


123.42 


15 




176.71 


1321-9 


28 


3 


626.80 


4688.8 


4 8 


17. 10 


127.95 


i5 


3 


182.65 


1366.4 


28 


6 


637-94 


4772-1 


4 9 


17.72 


132-56 


15 


6 


188.69 


1411-5 


28 


9 


649.18 


4856.2 • 


4 10 


i8.35 


137-25 


i5 


9 


194.83 


1457-4 


29 




660.52 


4941.0 


4 11 


18.99 


142 .02 


16 




201.06 


1504-1 


29 


3 


671.96 


5026.6 


5 


19.63 


146.88 


16 


3 


207.39 


i55i-4 


29 


6 


683 . 49 


5H2.9 


5 1 


20.29 


151.82 


16 


6 


213.82 


1599-5 


29 


9 


695.13 


5I99.9 


5 2 


20.97 


156.83 


16 


9 


220.35 


1648.4 


3° 




706.86 


5287.7 


5 3 


21.65 


161.93 


17 




226.98 


1697.9 










S * 


22 .34 


167. 12 


17 


3 233.71 


1748.2 











90 



PLATES AND TABLES 



91 



TABLE XII. — STANDARD DIMENSIONS OF WROUGHT-IRON 

WELDED PIPE. 

(National Tube Works.) 



Nominal 


Actual 


Actual 










Inside 


Outside 


Inside 


Internal Area. 


External Area. 


Diameter 


Diameter. 


Diameter. 










Ins. 


Ins. 


Ins. 


Sq. In. 


Sq. Ft. 


Sq. In. 


Sq. Ft. 


1 

8 


•405 


.270 


•057 


.0004 


. 1288 


.0009 


1 

4 


•540 


•3 6 4 


.104 


.0007 


.2290 


.0016 


t 


•675 


•493 


.191 


.0013 


•3578 


.0025 


1 
2" 


.840 


.622 


•3°4 


.0021 


•554 


.0038 


1 


I.050 


.824 


•533 


.0037 


.866 


.0060 


I 


I-3I5 


1.048 


.861 


.0060 


1-358 


.0094 


I* 


1 .660 


1.380 


1.496 


.0104 


2.164 


.0150 


I* • 


I.900 


1. 610 


2.036 


.0141 


2.835 


.0197 


2 


2 -375 


2.067 


3-35 6 


•0233 


4-43° 


.0308 


2§ 


2.875 


2.468 


4.780 


•0332 


6.492 


.0451 


3 ' 


3-5°° 


3.067 


7-383 


•°5I3 


9.621 


.0668 


3* 


4.000 


3-548 


9.887 


.0689 


12.566 


.0875 


4 


4.500 


4.026 


12.730 


.0884 


15.904 


. 1 104 


4* 


5.000 


4.508 


15.961 


.1108 


I9-635 


.1364 


5 


5-563 


5- o45 


19.986 


.1388 


24.301 


.1688 


6 


6.625 


6.065 


28.890 


.2006 


34-472 


• 2394 


7 


7.625 


7.023 


38.738 


.2690 


45 • 664 


• 3171 


8 


8.625 


7.981 


50.027 


•3474 


58.426 


•4057 


9 


9.625 


8-937 


62 . 730 


•4356 


72.760 


• 5053 


IO 


10.75 


10.018 


78.823 


•5474 


90.763 


• 6303 


ii 


n-75 


1 1 . 000 


95-°33 


.6600 


108.434 


• 7530 


12 


12.75 


1 2 . 000 


113.098 


•7854 


127.677 


.8867 


13 


14 


i3- 2 5 


137.887 


•9577 


I53-938 


I .0690 


14 


15 


14.25 


I59-485 


1. 1075 


176.715 


1.2272 


i5 


16 


15-25 


182.665 


1.2685 


201.062 


1-3963 


i7 


18 


17-25 


239.706 


1.6229 


254.470 


I. 7671 


i9 


20 


19.25 


291.040 


2.0211 


3I4.I59 


2.1817 


21 


22 


21.25 


354.657 


2.4629 


380.134 


2.6398 


23 


24 


23-25 


424.558 


2.9483 


45 2 -39° 


3-1416 



TABLE XIII. — HYPERBOLIC LOGARITHMS. 



N. 


Loga- 


N 


Loga- 


N. 


Loga- 


N. 


Loga- 




rithm. 


1 


ithm. 




ithm. 




rithm. 


I.OI 


.00995 


1-57 


.45108 


2.13 


.75612 


2.69 


.98954 


1. 02 


.01980 


1.58 


•45742 


2.14 


.76081 


2.70 


•99325 


1.03 


.02956 


1-59 


•46373 


2.15 


•76547 


2.71 


.99695 


1.04 


.03922 


I.60 


.47000 


2.16 


. 77011 


2.72 


1.00063 


1.05 


.04879 


I.61 


.47623 


2.17 


•77473 


2.73 


1.00430 


1.06 


.05827 


I.62 


.48243 


2.18 


77932 


2.74 


1.00796 


1.07 


.06766 


I.63 


.48858 


2.19 


78390 


2-75 


1.01160 


1.08 


.07696 


I.64 


.49470 


2.20 


.78846 


2.76 


1. 01523 


1.09 


.08618 


I.65 


50078 


2.21 


79299 


2.77 


1. 01885 


1. 10 


•°953i 


1.66 


50681 


2.22 


7975i 


2.78 


1.02245 


I. II 


• 10436 


1.67 


51282 


2.23 


80200 


2.79 


1.02604 


1. 12 


■**333 


1.68 


5i879 


2.24 


80648 


2.80 


1.02962 


I.I3 


. 12222 


1.69 


52473 


2.25 


81093 


2.8l 


1. 03318 


I.I4 


•i3 I0 3 


1.70 


53 o6 3 


2.26 


81536 


2.82 


1.03674 


I.I5 


•13977 


1 .71 


53649 


2.27 


81978 


2.83 


1.04028 


1. 16 


.14842 


1.72 


54232 


2.28 


82418 


2.84 


1.04380 


I.I7 


.15700 


i-73 


54812 


2.29 


82855 


2.85 


1.04732 


I.I8 


•16551 


1.74 


55389 


2.30 


83291 


2.86 


1.05082 


1. 19 


•17395 


i.75 


55962 


2.31 


83725 


2.87 


1-05431 


I.20 


. 18232 


1.76 


56531 


2.32 


84157 


2.88 


1.05779 


1 .21 


.19062 


1.77 


57098 


2-33 


84587 


2.89 


1. 06126 


1.22 


.19885 


1.78 


57661 


2-34 


8 5°i5 


2.90 


1 .06471 


1.23 


.20701 


1.79 


58222 


2-35 


85442 


2.91 


1. 06815 


1.24 


.21511 


1.80 


58779 


2.36 


85866 


2.92 


1. 07158 


1.25 


.22314 


1 .81 


59333 


2-37 


86289 


2-93 


1.07500 


1.26 


.23111 


1.82 


59884 


2.38 


86710 


2.94 


1. 07841 


1.27 


.23902 


1.83 . 


60432 


2.39 


87129 


2-95 


1.08181 


1,28 


.24686 


1.84 . 


60977 


2.4O 


87547 


2.96 


1. 085 19 


1.29 


.25464 


1.85 . 


61519 


2.4I 


87963 


2.97 


1.08856 


I.3O 


.26236 


1.86 


62058 


2.42 


88377 


2.98 


1. 09192 


I.3I 


.27003 


1.87 . 


62594 


2.43 


88789 


2-99 


1.09527 


1.32 


.27763 


1.88 


63127 


2.44 


89200 


3.00 


1. 09861 


1-33 


.28518 


1.89 • 


63658 


2.45 


89609 


3.01 


1. 10194 


i-34 


.29267 


1.90 


64185 


2.46 


90016 


3.02 


1. 10526 


i-35 


.30010 


1. 91 


64710 


2-47 


90422 


3.03 


1. 10856 


1.36 


.30748 


1.92 


65233 


2.48 


90826 


3.04 


1.11186 


i-37 


.31481 


1-93 


65752 


2.49 


91228 


3.05 


1.11514 


1.38 


.32208 


1.94 


66269 


2.50 


91629 


3-o6 


1. 11841 


1-39 


.32930 


1-95 


66783 


2.51 


92028 


3-07 


1.12168 


1.40 


•33647 


1.96 . 


67294 


2.52 


92426 


3.08 


1. 12493 


1. 41 


•34359 


1.97 


67803 


2-53 


92822 


3.09 


1. 12817 


1.42 


.35066 


1.98 


68310 


2.54 


93216 


3-io 


1.13140 


i-43 


•35767 


1.99 


68813 


2.55 


93609 


3-n 


1. 13462 


1.44 


.36464 


2.00 


69315 


2.56 


94001 


3-12 


I-I3783 


1-45 


•37156 


2.01 


69813 


2-57 


9439i 


3.13 


1 .14103 


1.46 


•37 8 44 


2.02 


70310 


2.58 


94779 


3-14 


1. 14422 


1.47 


.38526 


2.03 


70804 


2.59 


95166 


3-15 


1. 14740 


1.48 


.39204 


2.04 


71295 


2.6o 


95551 


3-i6 


I-I5057 


1.49 


•39878 


2.05 


71784 


2.6l 


95935 


3.17 


1 -15373 


1.50 


•40547 


2.06 


72271 


2.62 


96317 


3.18 


1. 15688 


i.5i 


.41211 


2.07 


72755 


2.63 


96698 


3-19 


1. 16002 


1.52 


.41871 


2.08 


73237 


2.64 


97078 


3.20 


1-16315 


1.53 


.42527 


2.09 


737i6 


2.65 


97454 


3.21 


1. 16627 


i-54 


•43178 


2.10 


74194 


2.66 


97833 


3.22 


1. 16938 


1. 55 


.43825 


2. 11 


74669 


2.67 


98208 


3.23 


1. 17248 


1.56 


■ 4446o 


2.12 


75142 


2.68 


98582 


3.24 


1 -17557 



92 



TABLE XIII. Continued. — HYPERBOLIC LOGARITHMS. 



N. 


Loga- 


N. 


Loga- 


N. 


Loga- 


N. 


Loga- 




rithm. 




rithm. 




rithm. 




rithm. 


3-25 


1. 17865 


3.8l 


I-33763 


4.37 


1.47476 


4-93 


!• 59534 


3-26 


1.18173 


3-82 


1-34025 


4.38 


I-47705 


4-94 


1 -59737 


3.27 


1. 18479 


3.83 


1.34286 


4-39 


1-47933 


4-95 


1-59939 


3.28 


I. 18784 


3.84 


1-34547 


4.40 


1 .48160 


4.96 


1.60141 


3-29 


1 19089 


3.85 


1.34807 


4.41 


1.48387 


4-97 


1 . 60342 


3-30 


1 . 19392 


3-86 


1 -35° 6 7 


4.42 


1. 48614 


4.98 


1 . 60543 


3.3i 


1 19695 


3.87 


i-353 2 5 


4.43 


1 . 48840 


4.99 


1 . 60744 


3-32 


1. 19996 


3-88 


I-35584 


4.44 


1 . 49065 


5.00 


1 . 60944 


3-33 


1 . 20297 


3-89 


1-35841 


4-45 


1 .49290 


5.oi 


1 .61144 


3-34 


1.20597 


3-90 


1.36098 


4.46 


I-495I5 


502 


1-61343 


3-35 


1 . 20896 


3.9i 


I-36354 


4-47 


1-49739 


5.03 


1 .61542 


3-36 


1 .21194 


3 -92 


1.36609 


4.48 


1.49962 


5-04 


1.61741 


3-37 


1.21491 


3-93 


1.36S64 


4-49 


1. 50185 


5.05 


1. 61939 


3-38 


1. 21788 


3-94 


i-37ii8 


4-5o 


1.50408 


5-o6 


1. 62137 


339 


1.22083 


3-95 


I-3737I 


4.5i 


1.50630 


5-07 


1.62334 


3 40 


1.22378 


396 


1.37624 


4.52 


1-50851 


5.08 


1. 62531 


3-41 


1. 22671 


3-97 


1.37877 


4-53- 


1. 51072 


5-09 


1.62728 


3 -42 


1.22964 


3.98 


1. 38128 


4-54 


1-51293 


5.10 


1 .62924 


3-43 


1.23256 


3-99 


I-38379 


4-55 


I-5I5I3 


5.ii 


1 . 63120 


3-44 


1-23547 


4.00 


1.38629 


4.56 


I-5I732 


5.12 


1-63315 


3-45 


1-23837 


4.01 


1.38879 


4.57 


-5I95I 


5.13 


1-63511 


3-46 


1. 24127 


4.02 


1. 39128 


4-58 


1. 52170 


5-14 


1 63705 


3-47 


1. 24415 


4-03 


1-39377 


4-59 


1.52388 


5.15 


1 . 63900 


348 


1.24703 


4.04 


1.39624 


4.60 


1.52606 


5.16 


1 . 64094 


3-49 


1 . 24990 


4.05 


1.39872 


4.61 


1.52823 


5.17 


1.64287 


3-50 


1.25276 


4.06 


1.40118 


4.62 


1 -53°39 


5.18 


1. 6448 1 


3.5i 


1.25562 


4.07 


1.40364 


4-63 


1-53256 


5.19 


1 . 64673 


3-52 


1.25846 


4.08 


1 .40610 


4.64 


I-5347I 


5-20 


1 . 64866 


3-53 


1. 26130 


4.09 


1 . 40854 


4.65 


1-53687 


5.21 


1 . 65058 


3.54 


1. 26412 


4.10 


1. 41099 


4.66 


1.53902 


5.22 


1.65250 


3-55 


1 .26695 


4.11 


1. 41342 


4.67 


i-54ii6 


5.23 


1. 65441 


3.56 


1.26976 


4.12 


1-41585 


4.68 


1 • 5433o 


5.24 


1.65632 


3-57 


1.27257 


4-13 


1. 41828 


4.69 


^- 54543 


5.25 


1.65823 


3.58 


I-2753 6 


4.14 


1 .42070 


4.70 


1 54756 


5.26 


1. 66013 


3-59 


1. 27815 


4-15 


1-42311 


4.71 


1 . 54969 


5.27 


1 . 66203 


3.60 


1.28093 


4.16 


1-42552 


4.72 


i-55i8i 


5-28 


1 . 66393 


3.61 


1. 28371 


4.17 


1.42792 


4-73 


1-55393 


5.29 


1.66582 


3.62 


1.28647 


4.18 


1 -4303 1 


4-74 


1.55604 


5.30 


1. 66771 


3.63 


1 .28923 


4.19 


1.43270 


4.75 


1-55814 


5.3i 


1 . 66959 


3.64 


1 .29198 


4.20 


1.43508 


4.76 


1.56025 


5.32 


1. 67147 


3.65 


1.29473 


4.21 


1.43746 


4-77 


1-56235 


5-33 


1-67335 


3-66 


1 .29746 


4.22 


1 . 43984 


4.78 


1.56444 


5-34 


1.67523 


3.67 


1. 30019 


4.23 


1 .44220 


4-79 


1-56653 


5-35 


1. 67710 


3.68 


1. 30291 


4.24 


1.44456 


4.80 


1.56862 


5.36 


1.67896 


3.69 


i-3°5 6 3 


4.25 


1 . 44692 


4.81 


1.57070 


5-37 


1 . 680S3 


3.70 


1-30833 


4.26 


1.44927 


4.82 


1.57277 


5.38 


1 .68269 


3.7i 


1.31103 


4.27 


1.45161 


4.83 


I-57485 


5-39 


1.68455 


3.72 


I-3I372 


4.28 


1-45395 


4.84 


1. 57691 


5.40 


1 . 68640 


3-73 


1.31641 


4.29 


1.45629 


4.85 


1.57898 


5-4i 


1.68825 


3-74 


1. 31909 


4.30 


1. 45861 


4.86 


1. 58104 


5-42 


1. 69010 


3-75 


1. 32176 


4-3i 


1 . 46094 


4.87 


1.58309 


5.43 


1 . 69194 


3-76 


1.32442 


4-32 


1 .46326 


4.88 


1-58515 


5-44 


1.69378 


3-77 


1.32707 


4-33 


I-46557 


4.89 


1. 58719 


5-45 


1.69562 


3.78 


1.32972 


4-34 


1.46787 


4.90 


1.58924 


5.46 


1.69745 


3-79 


I-33237 


4-35 


1 .47018 


4.91 


1. 59127 


5.47 


1 . 69928 


3.80 


1 • 335oo 


4.36 


1.47247 


4.92 


1. 5933 1 


5.48 


1 . 701 1 1 



93 



TABLE XIII Continued. — HYPERBOLIC LOGARITHMS. 



N. 


Loga- 


N. 


Loga- 


N. 


Loga- 


N. 


Loga- 




rithm. 




rithm. 




rithm. 




rithm. 


5-49 


1.70293 


6.05 


1.80006 


6.61 


1.88858 


7.17 


1 .96991 


5.50 


I-70475 


6.06 


1.80171 


6.62 


1. 89010 


7.18 


1. 97130 


5.51 


1 .70656 


6.07 


1.80336 


6.63 


1. 89 1 60 


7.19 


1.97269 


5-52 


1.70838 


6.08 


1 .80500 


6.64 


1-89311 


7.20 


1.97408 


5-53 


1 . 71019 


6.09 


1.80665 


6.65 


1.89462 


7.21 


1-97547 


5-54 


1.71199 


6.IO 


1 .80829 


6.66 


1. 89612 


7.22 


1.97685 


5-55 


1. 71380 


6. 11 


1 . 80993 


6.67 


1.89762 


7-23 


1.97824 


5.56 


1. 71560 


6.12 


1.81156 


6.68 


1. 8991 2 


7.24 


1.97962 


5.57 


1. 71740 


6.13 


1-81319 


6.69 


1 .90061 


7-25 


1. 98100 


5.58 


1.71919 


6.14 


1. 81482 


6.70 


1. 902 1 1 


7.26 


1.98238 


5-59 


1 . 72098 


6.15 


1. 81645 


6.71 


1.90360 


7-27 


1.98376 


5-6o 


1 . 72277 


6.16 


1. 81808 


6.72 


1.90509 


7.28 


1-98513 


5.61 


1-72455 


6.17 


1. 81970 


6-73 


1.90658 


7.29 


1.98650 


5-62 


1.72633 


6.18 


1 .82132 


6.74 


1 .90806 


7.30 


1.98787 


5.63 


1.72811 


6.19 


1 .82294 


6.75 


1.90954 


7-31 


1 .98924 


5-64 


1.72988 


6.20 


1-82455 


6.76 


1 .91102 


7-32 


1. 99061 


5.65 


1 .73166 


6.21 


1. 82616 


6.77 


1. 91250 


7-33 


1 .99198 


5-66 


1-73342 


6.22 


1.82777 


6.78 


1. 91398 


7-34 


1-99334 


5.67 


I-735I9 


6.23 


1.82937 


6.79 


I-9I545 


7-35 


1.99470 


5-68 


1 -73 6 95 


6.24 


1 . 83098 


6.80 


1. 9 1 692 


736 


1 .99606 


5 &9 


1. 73871 


6.25 


1.83258 


6.81 


1. 91839 


7-37 


1.99742 


5.7o 


1 . 74047 


6.26 


1. 83418 


6.82 


1. 91986 


7.38 


1.99877 


5.7i 


1 . 74222 


6.27 


I-83578 


6.83 


1 .92132 


7-39 


2.00013 


5.72 


1 - 74397 


6.28 


I-83737 


6.84 


1.92279 


7.40 


2.00148 


5-73 


1.74572 


6.29 


1.83896 


6.85 


1.92425 


7.41 


2.00283 


5-74 


1.74746 


6.30 


1.84055 


6.86 


1 -92571 


7.42 


2.00418 


5-75 


1 . 74920 


6.31 


1. 842 14 


6.87 


1 .92716 


7-43 


2.00553 


5.76 


1.75094 


6.32 


1.84372 


6.88 


1 .92862 


7-44 


2.00687 


5-77 


1.75267 


6-33 


1.84530 


6.89 


1.93007 


7-45 


2.00821 


5.78 


1 . 75440 


6-34 


1.84688 


6.90 


I-93I52 


7.46 


2.00956 


5-79 


i-756i3 


6.35 


1.84845 


6.91 


1.93297 


7-47 


2.01089 


5.80 


1.75786 


6.36 


1.85003 


6.92 


1.93442 


7.48 


2.01223 


5.81 


1-75958 


6-37 


1. 85160 


6-93 


1.93586 


7-49 


2-01357 


5.82 


1. 76130 


6.38 


I-853I7 


6.94 


I-9373Q 


7-50 


2.01490 


5.83 


1 . 76302 


6-39 


1-85473 


6-95 


r. 93874 


7-51 


2 .01624 


5.84 


1.76473 


6.40 


1.85630 


6.96 


1. 94018 


7-52 


2.01757 


5.85 


1 . 76644 


6.41 


1.85786 


6.97 


1 .94162 


7-53 


2.01890 


5-86 


1. 76815 


6.42 


■1.85942 


6.98 


I-94305 


7-54 


2 .02022 


5.87 


1.76985 


6-43 


1.86097 


6.99 


1 . 94448 


7-55 


2.02155 


5-88 


1. 77156 


6.44 


1.86253 


7.00 


1. 94591 


7-56 


2 .02287 


5-8g 


1.77326 


6-45 


1 . 86408 


7.01 


1-94734 


7-57 


2 .02419 


5-9o 


1-77495 


6.46 


1.86563 


7.02 


1.94876 


7.58 


2.02551 


5.9i 


1.77665 


6.47 


1. 86718 


7.03 


1. 95019 


7-59 


2.02683 


5-92 


I-77834 


6.48 


1.86872 


7.04 


1.95161 


7.60 


2.02815 


5-93 


1 . 78002 


6.49 


1 .87026 


7.05 


I-95303 


7.61 


2 .02946 


5-94 


1.78171 


6.50 


1. 87180 


7.06 


1-95444 


7.62 


2.03078 


5.95 


I-78339 


6.51 


1-87334 


7.07 


1.95586 


7.63 


2.03209 


5-96 


1.78507 


6.52 


1.87487 


7.08 


1.95727 


7.64 


2.03340 


5-97 


1.78675 


6.53 


1. 87641 


7.09 


1.95869 


7-65 


2.03471 


5-98 


1 . 78842 


6-54 


1.87794 


7.10 


1 . 96009 


7.66 


2 .03601 


5-99 


1 . 79009 


6.55 


1.87947 


7. 11 


1. 96150 


7.67 


2.03732 


6.00 


1 . 79176 


6.56 


1 . 88099 


7.12 


1. 96291 


7.68 


2 .03862 


6.01 


1.79342 


6.57 


1. 88251 


7-i3 


1. 9643 1 


7.69 


2.03992 


6.02 


1 . 79509 


6.58 


1 . 88403 


7.14 


1. 96571 


7.70 


2 .04122 


6.03 


I-79675 


6-59 


1-88555 


7-15 


1.96711 


7.71 


2.04252 


6.04 | 1.79840 


6.60 


1.88707 


7.16 


1. 9685 1 


7-72 


2 .04381 



94 



TABLE XIII. Continued.— HYPERBOLIC LOGARITHMS. 



N. 


Loga- 


N. 


Loga- 


N. 


Loga- 


N. 


Loga- 




rithm. 




rithm. 




rithm. 




rithm. 


7-73 


2. 0451 1 


8.30 


2.11626 


8.87 


2 . 18267 


9.44 


2 . 24496 


7-74 


2 . 04640 


8.31 


2. 1 1 746 


8.88 


2.18380 


9-45 


2 . 24601 


7-75 


2.04769 


8.32 


2.11866 


8.89 


2.18493 


9.46 


2 . 24707 


7.76 


2.04898 


8.33 


2. 11986 


8.90 


2. 18605 


9-47 


2 .24813 


7-77 


2.05027 


8.34 


2. 12106 


8.91 


2. 18717 


9.48 


2.24918 


7.78 


2.05156 


8-35 


2 . 12226 


8.92 


2.18830 


9.49 


2 .25024 


7-79 


2.05284 


8.36 


2. 12346 


8.93 


2 . 18942 


9-5o 


2 . 25129 


7.80 


2.05412 


8.37 


2 . 12465 


8.94 


2.19054 


9.5i 


2.25234 


7.81 


2.05540 


8.38 


2.12585 


8.95 


2.19165 


9-52 


2-25339 


7.82 


2.05668 


8.39 


2. 12704 


8.96 


2. 19277 


9-53 


2.25444 


7.83 


2.05796 


8.40 


2.12823 


8.97 


2.19389 


9.54 


2.25549 


7.84 


2.05924 


8.41 


2. 12942 


8.98 


2 . 19500 


9-55 


2.25654 


7.85 


2.06051 


8.42 


2. 13061 


8-99 


2 . 19611 


9.56 


2-25759 


7.86 


2.06179 


8-43 


2. 13180 


9.00 


2.19722 


9-57 


2.25863 


7.87 


2.06306 


8-44 


2. 13298 


9.01 


2.19834 


9.58 


2 .25968 


7.88 


2.06433 


8.45 


2-I34I7 


9.02 


2.19944 


9-59 


2.26072 


7.89 


2.06560 


8.46 


2-13535 


9-03 


2.20055 


9.60 


2 . 26176 


7.90 


2.06686 


8.47 


2-i3 6 53 


9.04 


2.20166 


9.61 


2 . 26280 


7.91 


2.06813 


8.48 


2-i377i 


9-05 


2 . 20276 


9.62 


2 . 26384 


7.92 


2.06939 


8-49 


2.13889 


9.06 


2 .20387 


9-63 


2.26488 


7-93 


2.07065 


8.50 


2. 14007 


9.07 


2 . 20497 


9.64 


2 . 26592 


7-94 


2 .07191 


8.51 


2. 14124 


9,08 


2 . 20607 


9.65 


2 .26696 


7-95 


2.07317 


8.52 


2. 14242 


9.09 


2 . 20717 


9.66 


2 . 26799 


7.96 


2.07443 


8.53 


2-14359 


9.10 


2 . 20827 


9.67 


2 . 26903 


7-97 


2.07568 


8-54 


2. 14476 


9. 11 


2.20937 


9.68 


2 . 27006 


7.98 


2.07694 


8.55 


2-14593 


9.12 


2 . 21047 


9.69 


2 . 27109 


7-99 


2.07819 


8.56 


2. 14710 


9-13 


2.21157 


9.70 


2 27213 


8.00 


2.07944 


8.57 


2.14827 


9.14 


2 . 21266 


9.71 


2 . 27316 


8.01 


2 .08069 


8.58 


2 - 14943 


9-15 


2.21375 


9.72 


2 . 27419 


8.02 


2.08194 


8.59 


2. 15060 


9.16 


2 . 21485 


9-73 


2.27521 


8.03 


2.08318 


8.60 


2.15176 


9.17 


2.21594 


9-74 


2 . 27624 


8.04 


2.08443 


8.61 


2 . 15292 


9.18 


2 . 21703 


9-75 


2 .27727 


8.05 


2.08567 


8.62 


2.15409 


9.19 


2 . 21812 


9.76 


2 . 27829 


8.06 


2.08691 


8.63 


2.15524 


9.20 


2 . 21920 


9-77 


2.27932 


8.07 


2.08815 


8.64 


2. 15640 


9.21 


2 . 22029 


9.78 


2 .28034 


8.08 


2 . 08939 


8.65 


2.15756 


9.22 


2 . 22138 


9-79 


2 .28136 


8.09 


2 . 09063 


8.66 


2.15871 


9-23 


2 . 22246 


9.80 


2 .28238 


8.10 


2.09186 


8.67 


2.15987 


9.24 


2.22351 


9.81 


2 . 28340 


8. 11 


2 .09310 


8.68 


2. 16102 


9-25 


2.22462 


9.82 


2 . 28442 


8.12 


2.09433 


8.69 


2. 1 62 1 7 


9.26 


2 . 22570 


9.83 


2.28544 


8.13 


2.09556 


8.70 


2.16332 


9.27 


2 .22678 


9.84 


2.28646 


8.14 


2.09679 


8.71 


2. 16447 


9.28 


2 . 22786 


9.85 


2.28747 


8.15 


2.09802 


8.72 


2 . 16562 


9.29 


2 .22894 


9.86 


2.28849 


8.16 


2.09924 


8.73 


2 . 16677 


9-30 


2 . 23001 


9.87 


2. 28950 


8.17 


2 . 10047 


8.74 


2 . 16791 


9-3i 


2. 23109 


9.88 


2.29051 


8.18 


2. 10169 


8-75 


2 . 16905 


9-32 


2 . 23216 


9.89 


2 .29152 


8.19 


2. 10291 


8.76 


2 . 17020 


9-33 


2.23323 


9.90 


2.29253 


8.20 


2.10413 


8.77 


2.17134 


9-34 


2.23431 


9.91 


2-29354 


8.21 


2.10535 


8.78 


2 . 17248 


9-35 


2.23538 


9.92 


2.29455 


8.22 


2. 10657 


8-79 


2. 17361 


9-36 


2.23645 


9-93 


2.29556 


8.23 


2. 10779 


8.80 


2-17475 


937 


2.23751 


9.94 


2.29657 


8.24 


2.10900 


8.81 


2.17589 


9.38 


2.23858 


9-95 


2.29757 


8.25 


2.11021 


8.82 


2. 17702 


9-39 


2.23965 


9.96 


2.29858 


8.26 


2.11142 


8.83 


2. 17816 


9.40 


2.24071 


9-97 


2.29958 


8.27 


2.11263 


8.84 


2 . 17929 


9.41 


2.24177 


9.98 


2.30058 


8.28 


2. 1 1384 


8.85 


2. 18042 


9.42 


2.24184 


9.99 


2.30158 


8.29 


2 . H-505 


8.86 


2. 1 81 55 


9-43 


2. 24390 







95 



LOGARITHMS OF NUMBERS. 



No. 



IOO OO 

IOI 

I02 



I03 
IO4 
I05 
IO6 
I07 
IO8 

IOg 
110 

III 

112 

113 
114 

115 
Il6 
117 

Il8 
IIQ 
I20 

121 
122 
123 
124 

125 
126 

127 
128 
129 

I30 

131 
132 

133 
134 
135 
I36 
137 
138 

139 
140 
141 
142 

143 
144 

145 
I46 

147 
I48 
149 



OI 



02 



°3 



04 



°5 



06 



07 188 



08 



09 



10 



11 



12 



13 



14 



IS 



16 



17 



000 

43 2 
860 

284 

7°3 
119 

53i 
938 
342 

743 
139 
53 2 
922 
308 
690 

070 
446 
819 

1 

555 
918 

279 
636 
991 

342 
691 

o37 
380 
721 
o59 

394 
727 

o57 

385 
710 

033 

354 
672 
988 

301 
613 
922 

229 

534 
836 

137 

435 
732 
026 
3*9 



°43 
475 
9°3 
326 

745 
160 

572 
979 
3*3 
782 
179 
57i 
961 
346 
729 

108 

483 
856 

225 

59i 
954 

3i4 
672 

*02 6 



087 

518 

945 
368 

787 
202 

612 
*oi9 
423 
822 
218 
610 

999 

385 
767 

145 
521 
893 
262 
628 
990 

35° 
707 

*o6i 



130 
56i 



377 
726 

072 

4i5 

755 
°93 
428 
760 
090 

418 

743 
066 

386 

704 

*oi9 

333 

644 

953 

259 
564 
866 

167 
465 
761 

056 
348 



412 
760 
106 

449 
789 
126 

461 

793 

123 

45° 
775 
098 

418 

735 
*o5i 

3 6 4 

675 

983 

290 

594 

897 

197 

495 
791 

085 

377 



410 
828 

243 

653 

*o6o 

463 
862 

258 

650 

#038 

423 
805 

183 

558 
93° 
298 
664 

*02 7 

386 

743 
#096 

447 
795 
140 

483 
823 
160 

494 

826 

156 

483 
808 
130 

45° 

767 
*o82 



395 

706 

*oi4 

320 
625 

927 
227 

524 
820 

114 

406 



173 
604 

*03<D 

452 
870 

284 

694 

*IOO 

5°3 
902 

297 
689 

*o77 

461 

843 

221 

595 
967 

335 

700 

^063 

422 

778 

*I X2 

482 
83O 

175 

517 
857 
193 
528 
860 
189 

516 
84O 
l62 
481 

799 
*H4 

426 

737 
*o45 

35i 
655 
957 
256 

554 
850 

143 
435 



217 
647 

+072 

494 
912 

325 

735 

*i4i 

543 
941 

33 6 

727 

*ii5 
500 
881 

258 

£>33 
*oo4 

372 

737 
*o99 

458 

814 

*i67 

5i7 
864 
209 

55i 
890 

227 

561 

893 
222 

548 
872 
194 

5i3 
830 

*i45 

457 
768 

+076 

381 
685 

987 

286 

584 

879 

173 
464 



260 

689 

*ii5 

53 6 
953 
366 

776 

*i8i 

583 

981 

376 
766 

*i54 
538 
918 

296 
670 

*c>4i 

408 

773 
**35 



493 
849 

*202 

552 
899 
243 

585 
924 

26l 

594 
926 

254 

58i 

9°5 
226 



545 
862 

*i76 

489 

799 
*io6 

412 

715 
*oi7 

316 
613 
909 

202 

493 



3°3 

13 2 

*i57 

578 

995 
407 

816 

*222 
623 

*02I 

415 
805 

*I92 

576 

95 6 

333 

707 
*c>78 

445 
809 

*i7i 

529 
884 

*237 

587 
934 
278 

619 

958 

294 

628 

959 
287 

613 

937 
258 

577 

893 

*2o8 



520 

829 

*i37 
442 
746 

*o47 

346 
643 
938 
231 
522 



346 

775 
*i99 

620 
*o36 

449 

857 

*262 
663 

*o6o 

454 
844 

*23I 

614 
994 

31 1 
744 
*"S 
482 
846 

*207 

565 

920 
^272 

621 

968 
312 

653 
992 

3 2 7 
661 

992 

320 

646 
969 
290 

609 

925 
*239 

55i 

860 

*i68 

473 

776 

*o77 

376 

673 
967 

260 

55i 



389 
817 

*242 
662 

+078 
490 
898 

*302 

7°3 

*IOO 

493 
883 

^269 

652 

+032 

408 

781 

*i 5 i 

5i8 
882 

*243 
600 
955 

*3°7 
656 

*oo3 
346 
687 

*025 

361 

694 

*024 

352 
678 

*OOI 

322 

640 

956 

*270 

582 
891 
198 

5°3 
806 
*io7 
406 
702 
997 
289 
580 



Pp. Pts. 



44 


43 


4-4 
8.8 


4-3 
8.6 


13.2 


12.9 


17.6 


17.2 


22.0 


21.5 


26.4 


25.8 


30.8 


30.1 


35-2 
39-6 


34-4 
38.7 



41 

4.1 
8.2 


40 

4.0 
8.0 


12.3 
16.4 


12.0 
16.0 


20.5 


20.0 


24.6 
28.7 


24.0 
28.0 


32.8 
36.9 


32.0 
36.0 



38 


37 


3-8 


3-7 


7.6 


7-4 


11. 4 


11. 1 


15-2 


14.8 


19.0 


18. 5 


22.8 


22.2 


26.6 


25-9 


3°-4 


29.6 


34-2 


33-3 



35 

3-5 

7.0 

10.5 

14.0 

17-5 
21.0 

24-5 
28.0 

31-5 



34 

3-4 
6.8 
10. 2 
13-6 
17.0 
20.4 
23.8 
27.2 
30.6 



32 


31 


3-2 


3-i 


6.4 


6.2 


9.6 


9-3 


12.8 


12.4 


16.0 


IS- 5 


19.2 


18.6 


22.4 


21.7 


25.6 


24.8 


28.8 


27.9 



42 

4.2 

8.4 

:2.6 
16.8 
21.0 

25.2 
29.4 
33-6 
J37.8 



39 

3-9 
7.8 
11. 7 
IS- 6 
19. S 
23-4 
27.3 
31.2 

35-i 



36 

3-6 
7.2 
10.8 
14.4 
18.0 
21.6 
25.2 
28.8 
32.4 



33 
3-3 

6.6 

9.9 

13 

16 

10 

23 

26.4 

29.7 



30 

3-o 
6.0 
9.0 
12.0 
iS-o 
18.0 
21.0 
24.0 
27.0 



96 



LOGARITHMS OF NUMBERS. 



No. 

I50 





1 


2 


3 


4 


5 


6 


7 


8 


9 


Pp. Pts. 


17 609 


638 


667 


696 


725 


754 


782 


8n 


840 


869 




151 


898 


926 


955 


984 


*oi3 


*c>4i 


*o7o 


*o99 


*I27 


*i 5 6 




152 


18 184 


213 


241 


270 


298 


327 


355 


384 


412 


44i 


29 


28 


153 


469 


498 


526 


554 


583 


611 


639 


667 


696 


724 


1 2.9 
258 


2.8 

<? 6 


154 


75 2 


780 


808 


837 


865 


893 


921 


949 


977 


*oo5 


3 8. 7 


- ^ 
8.4 


155 


19 °33 


061 


089 


117 


145 


173 


201 


229 


257 


285 


4 11. 6 


11. 2 


156 


312 


34o 


36S 


39 6 


424 


451 


479 


5o7 


535 


562 


5 14-5 
5 17.4 


14.0 
16.8 


157 


59° 


618 


645 


673 


700 


728 


756 


783 


811 


838 


7 20.3 


19.6 


158 


866 


893 


921 


948 


976 


*oo3 


*c>3o 


*o 5 8 


♦085 


*II2 


3 23.2 
9 26. 1 


22.4 
25.2 


159 


20 140 


167 


194 


222 


249 


276 


3°3 


33° 


358 


385 




160 


412 


439 


466 


493 


520 


548 


575 


602 


629 


656 




161 


683 


710 


737 


7 6 3 


790 


817 


844 


871 


898 


925 




162 


95 2 


978 


*oo5 


*032 


*°59 


♦085 


*II2 


*i 39 


*i65 


*I92 


27 

12 7 


26 
2.6 


163 


21 219 


245 


272 


299 


325 


352 


37S 


405 


43i 


458 


2 5-4 


5-2 


164 


484 


5" 


537 


5 6 4 


59° 


617 


643 


669 


696 


722 


3 8.1 


7.8 


165 


748 


775 


801 


827 


854 


880 


906 


932 


958 


985 


4 10.8 

5 13-5 


10.4 

13.0 


166 


22 Oil 


037 


063 


089 


"5 


141 


167 


194 


220 


246 


6 16.2 


15.6 


167 


272 


298 


324 


35° 


37 6 


401 


427 


453 


479 


505 


7 18.9 
? 21.6 


18.2 
20.8 


168 


53i 


557 


583 


608 


634 


660 


686 


712 


737 


763 


9|24-3 


23.4 


169 


789 


814 


840 


866 


891 


917 


943 


968 


994 


*oi9 




170 


23 °45 


070 


096 


121 


i47 


172 


198 


223 


249 


274 




171 


300 


325 


35° 


376 


401 


426 


452 


477 


502 


528 


25 
1 2. t 


172 


553 


578 


603 


629 


654 


679 


704 


729 


754 


779 


173 


805 


830 


855 


880 


9°5 


93° 


955 


980 


*°°5 


^030 


2 5 





174 


24 055 


080 


105 


130 


i55 


180 


204 


229 


254 


279 


3 7 

4 10 


5 



175 


3°4 


3 2 9 


353 


378 


403 


428 


452 


477 


502 


527 


5 12 


5 


176 


. 55i 


576 


601 


625 


650 


674 


699 


724 


74S 


773 


6 15 

7 17 



5 


177 


797 


822 


846 


871 


895 


920 


944 


969 


993 


*oi8 


8 20 





178 


25 042 


066 


091 


115 


i39 


164 


18S 


212 


237 


261 


9 22.5 


179 


285 


310 


334 


358 


382 


406 


43i 


455 


479 


5°3 




180 


527 


55i 


575 


600 


624 


648 


672 


696 


720 


744 




l8l 


768 


792 


816 


840 


864 


888 


912 


935 


959 


983 


24 


23 


182 


26 007 


031 


o55 


079 


102 


126 


i5° 


i74 


198 


221 


1 2.4 


2.3 


183 


245 


269 


293 


316 


34o 


3 6 4 


387 


411 


435 


458 


2 4.8 

3 7-2 


4.6 
6.9 


184 


482 


505 


5 2 9 


553 


576 


600 


623 


647 


670 


694 


4 9-6 


9.2 


185 


717 


74i 


764 


788 


811 


834 


858 


881 


905 


928 


5 12.0 


11. S 

,, 


186 


95i 


975 


998 


*02I 


*°45 


*o68 


*CX)I 


*ii4 


♦138 


*i6i 


14. 4 
7 16.8 


13. 

16. 1 


187 


27 184 


207 


231 


254 


277 


300 


323 


346 


37° 


393 


8 19.2 


18.4 


188 


416 


439 


462 


485 


508 


53 1 


554 


577 


600 


623 


921.6 


20.7 


189 


646 


669 


692 


715 


738 


761 


784 


807 


830 


852 




190 


875 


898 


921 


944 


967 


989 


*OI2 


*°35 


*o58 


*oSi 




191 


28 103 


126 


149 


171 


194 


217 


24O 


262 


285 


3°7 


22 


21 


192 


33° 


353 


375 


398 


421 


443 


466 


488 


5ii 


533 


1 2.2 
244 


2. I 

4 2 


193 


556 


578 


601 


623 


646 


668 


69I 


7i3 


735 


758 


3 6.6 


6-3 


194 


780 


803 


825 


847 


870 


892 


914 


937 


959 


981 


4 8.8 


8.4 


195 


29 003 


026 


048 


070 


092 


ii5 


137 


159 


181 


203 


5 11. 

6 13.2 


10.5 
12.6 


196 


226 


248 


270 


292 


3*4 


33 6 


358 


380 


403 


425 


7 iS-4 


14.7 


197 


447 


469 


491 


513 


535 


557 


579 


601 


623 


6 45 


017.6 
9 19.8 


16.8 
18.0 


198 


667 


688 


710 


732 


754 


776 


798 


820 


842 


863 




299 


885 
3« 


907 


929 


951 


973 


994 


*oi6 


*o38 


*o6o 


*o8i 





97 



■LOGARITHMS OF NUMBERS. 



No. 


O 


I 


2 


3 


4 


5 


6 


7 


8 


9 


Pp. Pts. 


200 


3° 103 


125 


146 


168 


190 


211 


233 


255 


276 


298 




20I 


320 


34i 


3 6 3 


384 


406 


428 


449 


471 


492 


5i4 




202 


535 


557 


578 


600 


621 


643 


664 


685 


707 


728 


22 


21 


203 


75o 


771 


792 


814 


835 


856 


878 


899 


920 


942 


1 2. 

2 4.. 

3 6. 


22.1 

i 4-2 

5 6.3 


204 


9 6 3 


984 


*oo6 


*02 7 


^048 


*o69 


*(X)I 


*II2 


*i33 


*i 5 4 


205 


3i 175 


197 


218 


239 


260 


281 


302 


323 


345 


366 


4 8. 


i 8.4 


206 


387 


408 


429 


45° 


471 


492 


513 


534 


555 


576 


5 11. 

6 13. 


3 IO.5 
2 12.6 


207 


597 


618 


639 


660 


6S1 


702 


723 


744 


7 6 5 


785 


7 IS- 


X 14-7 


208 


806 


827 


848 


869 


890 


911 


93i 


952 


973 


994 


8 17. 

9 19. 


5 16.8 
3 18.0 


209 


3 2 OI 5 


°35 


056 


077 


098 


118 


139 


160 


181 


201 




2IO 


222 


243 


263 


284 


3°5 


3 2 5 


346 


366 


387 


408 




211 


428 


449 


469 


490 


5i° 


53i 


55*2 


572 


593 


613 




212 


634 


654 


6 75 


69.5 


7i5 


736 


756 


777 


797 


818 


j 


20 


213 


838 


858 


879 


899 


919 


940 


960 


980 


*OOI 


*02I 


2 


4.0 


214 


33 041 


062 


082 


102 


122 


143 


i°3. 


183 


203 


224 


3 


6.0 


215 


244 


264 


284 


3°4 


325 


345 


3 6 5 


385 


405 


425 


4 
5 


8. 
10. 


2l6 


445 


465 


486 


506 


526 


546 


566 


586 


606 


626 


6 


12.0 


217 


646 


666 


686 


706 


726 


746 


766 


786 


806 


826 


7 
8 


14.0 
16. 


2l8 


846 


866 


885 


9°5 


925 


945 


965 


985 


*oo5 


*025 


9 


18.0 


2IQ 


34 044 


064 


084 


104 


124 


143 


163 


183 


203 


223 




220 


242 


262 


282 


301 


321 


34i 


361 


380 


400 


420 




221 


439 


459 


479 


498 


5i8 


537 


557 


577 


596 


6l6 




t n 


222 


635 


655 


674 


694 


7*3 


733 


753 


772 


792 


8ll 


1 


1.9 


223 


830 


850 


869 


889 


908 


928 


947 


967 


986 


*oo5 


2 


3-8 


224 


35 ° 2 5 


044 


064 


083 


102 


122 


141 


160 


180 


199 


3 

4 


5-7 
7.6 


225 


218 


238 


257 


276 


295 


3i5 


334 


353 


372 


392 


5 


9-5 


226 


411 


43° 


449 


468 


488 


5°7 


526 


545 


5 6 4 


583 


6 

7 


11. 4 
13 • 3 


227 


603 


622 


641 


660 


679 


698 


717 


736 


755 


774 


8 


15.2 


228 


793 


813 


832 


851 


870 


889 


908 


927 


946 


965 


9 


17. 1 


229 


984 


*oc>3 


*02I 


*040 


*o59 


+078 


*o97 


*n6 


**35 


*i54 




23O 


3 6 173 


192 


211 


229 


248 


267 


286 


3°5 


324 


342 




231 


361 


380 


399 


418 


43 6 


455 


474 


493 


5ii 


53° 




18 


232 


549 


568 


586 


605 


624 


642 


661 


680 


698 


717 


1 


1.8 


233 


736 


754 


773 


791 


810 


829 


847 


866 


884 


903 


2 
3 


3-6 

5-4 


234 


922 


940 


959 


977 


996 


*oi4 


*o 33 


*o5i 


*o7o 


*o88 


4 


7.2 


235 


37 107 


125 


144 


162 


181 


199 


218 


236 


254 


273 


5 
6 


9.o 

10.8 


236 


291 


3 IQ 


328 


346 


3 6 5 


383 


401 


420 


438 


457 


7 


12. 6 


237 


475 


493 


5ii 


53° 


548 


566 


585 


603 


621 


639 


8 


14.4 
16. 2 


238 


658 


676 


694 


712 


73 1 


749 


767 


785 


803 


822 


9 


239 


840 


8 S 8 


876 


894 


912 


93i 


949 


967 


985 


*oo3 




24O 


38 021 


°39 


°57 


o75 


°93 


112 


130 


148 


166 


184 




24I 


202 


220 


238 


256 


274 


292 


310 


328 


346 


3 6 4 




17 


242 


382 


399 


417 


435 


453 


471 


489 


507 


525 


543 


I 
2 


1-7 

3.4 


243 


56i 


578 


596 


614 


632 


650 


668 


686 


7°3 


721 


3 


5-i 


244 


739 


757 


775 


792 


810 


828 


846 


863 


881 


899 


4 
5 
6 


6.8 
8 c 


245 


917 


934 


952 


970 


987 


*oo5 


*023 


*04i 


*o58 


*o76 


. j 

EO. 2 


246 


39 °94 


in 


129 


146 


164 


182 


199 


217 


235 


252 


7 
8 
9 


EI. 9 

C3-6 
IS. 3 


247 


270 


287 


3°5 


322 


34o 


358 


375 


393 


410 


428 


248 


445 


463 


480 


498 


5i5 


533 


55o 


568 


585 


602 




249 


620 


637 


655 


672 


690 


707 


724 


742 


759 


777 





98 



LOGARITHMS OF NUMBERS. 



No. 





1 


2 


3 


4 


5 


6 


7 


8 


9 


Pp. Pts. 


250 


39 794 


811 


829 


846 


863 


881 


898 


9i5 


933 


95° 




251 


967 


985 


*002 


*oi9 


*°37 


*o54 


*o7i 


*o88 


*ro6 


*I23 




252 


40 140 


157 


175 


192 


209 


226 


2 43 


261 


278 


2 95 




18 


253 


312 


3 2 9 


346 


3 6 4 


381 


398 


4i5 


43 2 


449 


466 


I 


1.8 
3-6 

5.4 


254 


483 


500 


518 


535 


552 


569 


586 


603 


620 


637 


2 

3 


255 


654 


671 


688 


7°5 


722 


739 


75 6 


773 


790 


807 


4 


7.2 


256 


• 824 


841 


858 


875 


892 


909 


926 


943 


960 


976 


5 
6 


9.0 
10.8 


257 


993 


*OIO 


*02 7 


*044 


*o6i 


*o78 


*o 9 5 


*ni 


*I28 


*i 4 5 


7 


12.6 


258 


41 162 


179 


196 


212 


229 


246 


263 


280 


296 


3*3 


8 



14.4 


259 


33° 


347 


3 6 3 


380 


397 


414 


43° 


447 


464 


481 


y »»• • * 


260 


497 


514 


53i 


547 


5 6 4 


58i 


597 


614 


631 


647 




261 


664 


681 


697 


714 


73i 


747 


764 


780 


797 


814 




262 


830 


847 


863 


880 


896 


9i3 


929 


946 


963 


979 




17 


263 


996 


*OI2 


*029 


*<H5 


*o62 


♦078 


*°95 


*iii 


*I27 


*I44 


1 
2 


1.7 
3.4 


264 


42 160 


177 


193 


210 


226 


243 


2 59 


2 75 


292 


308 


3 


5-i 


265 


3 2 5 


341 


357 


374 


390 


406 


4 2 3 


439 


455 


472 


4 
5 


6.8 
8.5 


266 


488 


5°4 


521 


537 


553 


57° 


586 


602 


619 


635 


6 


10. 2 


267 


651 


667 


684 


700 


716 


732 


749 


765 


781 


797 


7 
8 


11. 9 


268 


813 


830 


846 


862 


878 


894 


911 


927 


943 


959 


9 


13. 
15.3 


269 


975 


991 


*oo8 


*024 


*040 


*c>56 


+072 


*o88 


*io4 


*I20 




270 


43 *3 6 


*5 2 


169 


185 


201 


217 


233 


249 


265 


28l 




271 


297 


3*3 


3 2 9 


345 


361 


377 


393 


409 


4 2 5 


441 




16 

1.6 


272 


457 


473 


489 


505 


52i 


537 


553 


569 


584 


60O 


1 


273 


616 


632 


648 


664 


680 


696 


712 


727 


743 


759 


2 


3-2 


274 


775 


791 


807 


823 


838 


854 


870 


886 


902 


917 


3 

4 


4.8 
6.4 


275 


933 


949 


965 


981 


996 


*OI2 


*028 


*044 


*o59 


*°75 


5 


8.0 


276 


44 091 


107 


122 


138 


154 


170 


185 


201 


217 


232 


6 

l-T 


9.6 


277 


248 


264 


2 79 


2 95 


3ii 


326 


342 


358 


373 


389 


7 

8 


12^8 


278 


404 


420 


436 


45i 


467 


483 


498 


514 


5 2 9 


545 


9 


14.4 


279 


560 


576 


592 


607 


623 


638 


654 


669 


685 


700 




280 


716 


73i 


747 


762 


778 


793 


809 


824 


840 


855 




28l 


871 


886 


902 


917 


932 


948 


963 


979 


994 


*OIO 




TCf 


282 


45 ° 2 5 


040 


056 


071 


086 


102 


117 


i33 


148 


163 


1 


1.5 


283 


179 


194 


209 


225 


240 


2 55 


271 


286 


301 


3 J 7 


2 
3 

4 


3 

A 




5 



284 


33 2 


347 


362 


378 


393 


408 


423 


439 


454 


469 


4 

6 


285 


484 


500 


5i5 


53° 


545 


56i 


576 


59i 


606 


621 


5 
6 

7 


7 


5 


286 


637 


652 


667 


682 


697 


712 


728 


743 


758 


773 


9 

10 




5 


287 


788 


803 


818 


834 


849 


864 


879 


894 


9°9 


924 


8 


12 





288 


939 


954 


969 


984 


*ooo 


*oi5 


^030 


*°45 


*o6o 


*o 7 5 


9 


13-5 


289 


46 090 


i°5 


120 


i35 


15° 


165 


180 


195 


210 


225 




29O 


240 


2 55 


270 


285 


300 


3i5 


33° 


345 


359 


374 




291 


389 


404 


419 


434 


449 


464 


479 


494 


5°9 


5 2 3 




14 


292 


538 


553 


568 


583 


598 


613 


627 


642 


657 


672 


1 
2 


1 
2 


4 
8 


293 


687 


702 


716 


73* 


746 


761 


776 


790 


805 


820 


3 


4 


2 


294 


835 


850 


864 


879 


894 


909 


923 


938 


953 


967 


4 


5 


6 


295 


982 


997 


*OI2 


*026 


*04i 


*o56 


+070 


*o85 


*IOO 


*ii4 


5 
6 


7 

8 



4 


296 


47 129 


144 


159 


173 


188 


202 


217 


232 


246 


261 


7 


9 


8 


297 


276 


290 


3°5 


319 


334 


349 


3 6 3 


378 


392 


407 


8 
9 


11 

12 


2 

6 


298 


422 


43 6 


45i 


465 


480 


494 


5°9 


524 


538 


553 




299 


567 


582 


596 


611 


625 


640 


654 


669 


683 


698 





99 



LOGARITHMS OF NUMBERS. 



No. 





1 


2 


3 


4 


5 

784 


6 


7 


8 

828 


9 

842 


Pp. Pts. 


300 


47 712 


727 


741 


756 


770 


799 


813 




301 


857 


871 


885 


900 


914 


929 


943 


958 


972 


986 




3°2 


48 001 


015 


029 


044 


058 


o73 


087 


IOI 


116 


130 




303 


144 


159 


173 


187 


202 


216 


230 


244 


259 


273 




304 


287 


302 


316 


33° 


344 


359 


373 


387 


401 


416 




305 


43° 


444 


458 


473 


487 


5°i 


5i5 


53° 


544 


558 




IS 


306 


57 2 


586 


601 


6i5 


629 


643 


657 


671 


686 


700 


1 

2 


I . 

3- 


3 
O 


307 


714 


728 


742 


75 6 


770 


785 


799 


813 


827 


841 


3 


4- 


5 


308 


855 


869 


883 


897 


911 


926 


940 


954 


968 


982 


4 
5 


6. 

7. 




5 


309 


996 


*OIO 


*024 


*o38 


*0 5 2 


*o66 


*o8o 


*cx)4 


*io8 


*I22 


6 


9- 





3IO 


49 *3 6 


i5° 


164 


178 


192 


206 


220 


234 


248 


262 


7 
g 


10 
12 


5 


311 


276 


290 


3°4 


3i8 


332 


346 


360 


374 


388 


402 


9 


13 


5 


312 


415 


429 


443 


457 


471 


485 


499 


5i3 


5 2 7 


541 




313 


554 


568 


582 


596 


6lO 


624 


638 


651 


665 


679 




314 


6 93 


707 


721 


734 


748 


762 


776 


790 


803 


817 




315 


831 


845 


859 


872 


886 


900 


914 


927 


941 


955 




316 


969 


982 


996 


*OIO 


*024 


*°37 


♦051 


+065 


*o79 


*092 




14 


317 


50 106 


120 


133 


147 


161 


174 


188 


202 


2 i5 


229 


1 
2 


1 
2 


4 
8 


3I§ 


243 


256 


270 


284 


297 


311 


325 


338 


35 2 


3 6 5 


3 


4 


2 


319 


379 


393 


406 


420 


433 


447 


461 


474 


4S8 


5°i 


4 
5 
6 


5 


6 


320 


5i5 


529 


542 


556 


5 6 9 


583 


596 


610 


623 


6 37 


7 
8 


4 


321 


651 


664 


678 


691 


7o5 


718 


73 2 


745 


759 


772 


7 
g 


9 


8 


322 


786 


799 


813 


826 


840 


853 


866 


880 


893 


907 


9 


12 


6 


323 


920 


934 


947 


961 


974 


987 


*OOI 


*oi4 


*028 


*04i 




324 


5i oss 


068 


081 


°95 


108 


121 


135 


148 


162 


175 




325 


188 


202 


215 


228 


242 


255 


268 


282 


2 95 


308 




326 


322 


335 


348 


362 


375 


388 


402 


415 


428 


441 




327 


455 


468 


481 


495 


508 


521 


534 


548 


5 6 * 


574 




13 


328 


587 


601 


614 


627 


640 


654 


667 


680 


693 


706 


1 


1 


3 
6 


329 


720 


733 


746 


759 


772 


786 


799 


812 


8 2 5 


838 


3 


3 


9 


330 


851 


865 


878 


891 


904 


917 


93° 


943 


957 


970 


4 
5 
6 


5 
6 


2 


331 


983 


996 


*oo9 


*022 


*°35 


+048 


*o6i 


*°75 


*o88 


*IOI 


7 


8 


332 


52 114 


127 


140 


153 


166 


179 


192 


205 


218 


231 


7 
8 

9 


9 


1 


333 


244 


257 


270 


284 


297 


310 


3 2 3 


33 6 


349 


362 


10 
11 


4 
7 


334 


375 


388 


401 


414 


427 


440 


453 


466 


479 


492 




335 


5°4 


5i7 


53° 


543 


556 


5 6 9 


582 


595 


608 


621 




336 


634 


647 


660 


673 


686 


699 


711 


724 


737 


750 




337 


7 6 3 


776 


789 


802 


815 


827 


840 


853 


866 


879 




338 


892 


9°5 


917 


93° 


943 


95 6 


969 


982 


994 


*oo7 




12 


339 


53 °2o 


°33 


046 


058 


071 


084 


097 


no 


122 


135 


1 


I 


2 


340 


148 


161 


173 


186 


199 


212 


224 


2 37 


250 


263 


3 


3 


4 
6 


341 


275 


288 


301 


3*4 


326 


339 


352 


3 6 4 


377 


39° 


4 


4 
< 


8 


342 


403 


4i5 


428 


441 


453 


466 


479 


491 


5°4 


5i7 


5 
6 




7 



2 


343 


5 2 9 


542 


555 


567 


580 


593 


605 


618 


631 


643 


7 


8 


• 4 


344 


656 


668 


681 


694 


706 


719 


732 


744 


757 


769 


8 

Q 


9 

TO 


.6 


345 


782 


794 


807 


820 


832 


845 


857 


870 


882 


895 


V -—•• — 


346 


908 


920 


933 


945 


958 


970 


983 


995 


*oo8 


*020 




347 


54 033 


045 


058 


070 


083 


°95 


108 


120 


133 


145 




348 


158 


170 


183 


195 


208 


220 


233 


2 45 


258 


270 




349 


283 


295 


3°7 


320 

1 


33 2 


345 


357 


37° 

1 


382 


394 





IOO 



LOGARITHMS OF NUMBERS. 



No. 





1 


2 


3 


4 


5 


6 


7 


8 


9 

5i8 


Pp. Pts. 


350 


54 407 


419 


43 2 


444 


45 6 


469 


481 


494 


506 




351 


53i 


543 


555 


568 


580 


593 


605 


617 


630 


642 




352 


654 


667 


679 


691 


704 


716 


728 


741 


753 


765 




353 


777 


790 


802 


814 


827 


839 


851 


864 


876 


888 




354 


900 


9i3 


9 2 5 


937 


949 


962 


974 


986 


998 


*on 




355 


55 ° 2 3 


°35 


047 


060 


072 


084 


096 


108 


121 


*33 




13 


356 


145 


157 


169 


182 


194 


206 


218 


230 


242 


2 55 


1 

2 


1 
2 


6 


357 


267 


2 79 


291 


3°3 


315 


328 


34o 


352 


3 6 4 


376 


3 


3 


9 


358 


388 


400 


4i3 


425 


437 


449 


461 


473 


485 


497 


4 

5 


5 
6 


2 

5 


359 


5°9 


522 


534 


546 


558 


57° 


582 


594 


606 


618 


6 


7 


8 


360 


630 


642 


654 


666 


678 


691 


7°3 


7i5 


727 


739 


7 
g 


9 


1 

A 


36i v 


75i 


763 


775 


787 


799 


811 


823 


835 


847 


859 


9 


11 


4 

7 


362 


871 


883 


895 


907 


919 


93i 


943 


955 


967 


979 




363 


991 


*oc>3 


*oi5 


*02 7 


*o38 


*o5o 


*o62 


*o74 


*o86 


*o98 




364 


56 no 


122 


i34 


146 


158 


170 


182 


194 


205 


217 




365 


229 


241 


2 53 


265 


277 


289 


301 


312 


324 


33 6 




366 


348 


360 


37 2 


384 


396 


407 


419 


43i 


443 


455 




12 


367 


467 


478 


490 


502 


5i4 


526 


538 


549 


56i 


573 


X 

2 


I 
2 


2 

4 


368 


585 


597 


608 


620 


632 


644 


656 


667 


679 


691 


3 


3 


6 


369 


7°3 


714 


726 


738 


75° 


761 


773 


785 


797 


808 


4 
5 
6 


4 
6 


8 


370 


820 


832 


844 


855 


867 


879 


891 


902 


914 


926 


7 


2 


37i 


937 


949 


961 


972 


984 


996 


*oo8 


*oi9 


*03i 


*°43 


7 
8 


8 


4 
6 


372 


57 °54 


066 


078 


089 


IOI 


113 


124 


136 


148 


159 


9 


9 

10 


8 


373 


171 


183 


194 


206 


217 


229 


241 


252 


264 


276 




374 


287 


299 


310 


322 


334 


345 


357 


368 


380 


392 




375 


403 


4i5 


426 


438 


449 


461 


473 


484 


496 


5°7 




376 


5i9 


53° 


542 


553 


565 


576 


588 


600 


611 


623 




377 


634 


646 


657 


669 


680 


692 


7°3 


7i5 


726 


738 




II 


378 


749 


761 


772 


784 


795 


807 


818 


830 


841 


852 


I 


I 


1 


379 


864 


875 


887 


898 


910 


921 


933 


944 


955 


967 


2 

3 


3 


3 


380 


978 


990 


*OOI 


*oi3 


*024 


*°35 


*o47 


#058 


^070 


*o8i 


4 

5 

6 


4 


4 


381 


58 092 


104 


ii5 


127 


138 


149 


161 


172 


184 


195 


5 
6 


5 
6 


382 


206 


, 218 


229 


240 


252 


263 


2 74 


286 


297 


3°9 


7 



7 


7 


383 


320 


33* 


343 


354 


365 


377 


388 


399 


410 


422 



9 


8 



8 



384 


433 


444 


45 6 


467 


478 


490 


5°i 


512 


524 


535 




385 


546 


557 


569 


580 


591 


602 


614 


625 


636 


647 




386 


659 


670 


681 


692 


704 


7i5 


726 


737 


749 


760 




387 


771 


782 


794 


805 


816 


827 


838 


850. 


861 


872 




388 


883 


894 


906 


917 


928 


939 


95° 


961 


973 


984 




10 


389 


995 


*oo6 


*6i7 


*o 2 8 


*040 


4=051 


*o62 


--H073 


*o8 4 


*°95 


I 


1 





390 


59 i° 6 


118 


129 


140 


151 


162 


173 


184 


195 


207 


2 
3 


2 
3 





39i 


218 


229 


240 


251 


262 


273 


284 


295 


306 


318 


4 


4 





392 


3 2 9 


34o 


35i 


362 


373 


384 


395 


406 


417 


428 


5 
6 


5 
6 






393 


439 


45° 


461 


472 


483 


494 


506 


5i7 


528 


539 


7 


7 





394 


55o 


56i 


572 


583 


594 


605 


616 


627 


638 


649 


8 
9 


8 





n 


395 


660 


671 


682 


693 


704 


715 


726 


737 


748 


759 


y • ~ 


396 


770 


780 


791 


802 


813 


824 


835 


846 


857 


868 




397 


879 


890 


901 


912 


923 


934 


945 


956 


966 


977 




398 


988 


999 


*OIO 


*02I 


*032 


*043 


*Q54 


*o65 


+076 


*o86 




399 


60 097 


108 


119 


I30 


141 


152 


163 


i73 


184 


195 





IOI 



LOGARITHMS OF NUMBERS. 



No. 





1 


2 


3 


4 


5 


6 


7 


8 


9 


Pp. Pts. 


40O 


60 206 


217 


228 


239 


249 


260 


271 


282 


293 


3°4 




4OI 


3i4 


325 


33 6 


347 


358 


3 6 9 


379 


39o 


401 


412 




402 


423 


433 


444 


455 


466 


477 


487 


498 


5°9 


520 




403 


53i 


54i 


552 


563 


574 


584 


595 


606 


617 


627 




404 


638 


649 


660 


670 


681 


692 


7°3 


7i3 


724 


735 




405 


746 


75 6 


767 


778 


788 


799 


810 


821 


831 


842 




406 


853 


863 


874 


885 


895 


906 


917 


927 


938 


949 




407 


959 


970 


981 


991 


*002 


*oi3 


*023 


*°34 


*Q45 


*°55 




408 


61 066 


077 


087 


098 


IO9 


119 


130 


140 


151 


162 


I 


11 

1.1 


409 


172 


183 


194 


204 


215 


225 


236 


247 


257 


268 


2 


2.2 | 


41O 


278 


289 


300 


310 


321 


33* 


342 


352 


3 6 3 


374 


3 
4 
5 


3-3 

A A. 


411 


384 


395 


405 


416 


426 


437 


448 


458 


469 


479 


-4 . if 

5-5 


412 


490 


500 


5ii 


52i 


532 


542 


553 


563 


574 


584 


6 

7 
8 


6.6 

7-7 
8.8 


413 


595 


606 


616 


627 


637 


648 


658 


669 


679 


690 


414 


700 


711 


721 


73i 


742 


752 


7 6 3 


773 


784 


794 


9 


9.9 


415 


805 


815 


826 


836 


847 


857 


86S 


878 


888 


899 




416 


909 


920 


93° 


941 


951 


962 


972 


982 


993 


*oo3 




417 


62 014 


024 


034 


045 


°55 


066 


076 


086 


097 


107 




418 


118 


128 


138 


149 


159 


170 


180 


190 


201 


211 




419 


221 


232 


242 


252 


263 


273 


284 


294 


3°4 


3i5 




420 


3 2 5 


335 


346 


35 6 


366 


377 


38/ 


397 


408 


418 




421 


428 


439 


449 


459 


469 


480 


490 


500 


5ii 


521 




10 


422 


53i 


542 


552 


562 


572 


583 


593 


603 


613 


624 


I 


1.0 


423 


634 


644 


655 


665 


675 


685 


696 


706 


716 


726 


2 


2.0 


424 


737 


747 


757 


767 


778 


788 


798 


808 


818 


829 


3 

4 


3-° 
4-0 


425 


839 


849 


859 


870 


880 


890 


900 


910 


921 


93i 


5 


5-o 


426 


941 


951. 


961 


972 


982 


992 


*002 


*OI2 


*022 


*o33 


6 

7 


6.0 
7.0 


427 


63 043 


•o53 


063 


o73 


083 


094 


IO4 


114 


124 


134 


8 


8.0 


428 


144 


155 


165 


175 


185 


i95 


205 


215 


225 


236 


9 


9.0 


429 


246 


256 


266 


276 


286 


296 


306 


3*7 


327 


337 




430 


347 


357 


367 


377 


387 


397 


407 


417 


428 


438 




431 


448 


458 


468 


478 


488 


498 


508 


5i8 


528 


538 




432 


548 


558 


568 


579 


589 


599 


609 


619 


629 


639 




433 


649 


659 


669 


679 


689 


699 


709 


719 


729 


739 




434 


749 


759 


769 


779 


789 


799 


809 


819 


829 


839 




435 


849 


859 


869 


879 


889 


899 


909 


919 


929 


939 




g 


436 


949 


959 


969 


979 


988 


998 


*oo8 


*oi8 


*028 


*o38 


1 


0.9 


437 


64 048 


058 


068 


078 


088 


098 


108 


118 


128 


137 


2 
3 

4 


i.8 

2.7 
3-6 


438 


147 


157 


167 


177 


187 


197 


207 


217 


227 


237 


439 


246 


256 


266 


276 


286 


296 


306 


316 


326 


335 


5 
6 


4-5 
5-4 
6-3 


440 


345 


355 


365 


375 


385 


395 


404 


414 


424 


434 


7 


441 


444 


454 


464 


473 


483 


493 


5°3 


5*3 


523 


532 


8 


7.2 
8.1 


442 


542 


552 


562 


572 


582 


59i 


601 


611 


621 


631 


9 


443 


640 


650 


660 


670 


680 


689 


699 


709 


719 


729 




444 


738 


748 


758 


768 


777 


787 


797 


807 


816 


826 




445 


836 


846 


856 


865 


875 


885 


895 


904 


914 


924 




446 


933 


943 


953 


9 6 3 


972 


982 


992 


*002 


*OII 


*02I 




447 


65 031 


040 


050 


060 


070 


079 


089 


O99 


108 


Il8 




448 


128 


137 


147 


157 


167 


176 


186 


I96 


205 


215 




449 


225 


234 


244 


254 


263 


273 


283 


292 


302 


312 





102 



LOGARITHMS OF NUMBERS 



No. 





1 


2 


3 


4 


5 


6 


7 


8 


9 


Pp.Pts. 


450 


65 3 21 


33i 


34i 


35° 


360 


3 6 9 


379 


389 


398 


408 




451 


418 


427 


437 


447 


45 6 


466 


475 


485 


495 


5°4 




452 


514 


523 


533 


543 


552 


562 


57i 


581 


59i 


600 




453 


610 


619 


629 


639 


648 


658 


667 


677 


686 


696 




454 


706 


7i5 


725 


734 


744 


753 


763 


772 


782 


792 




455 


801 


811 


820 


830 


839 


849 


858 


868 


877 


887 




456 


896 


906 


916 


925 


935 


944 


954 


9 6 3 


973 


982 




457 


992 


*OOI 


*OII 


*020 


^030 


*°39 


*049 


£058 


*o68 


*o77 




458 


66 087 


096 


106 


115 


124 


134 


143 


153 


162 


172 


I 


10 

1 . 


459 


181 


191 


200 


2IO 


219 


229 


238 


247 


257 


266 


2 


2.0 


460 


276 


285 


295 


3°4 


3i4 


323 


332 


342 


35i 


361 


3 


3° 


461 


37° 


380 


389 


398 


408 


417 


427 


43 6 


445 


455 


4 
5 


4.0 
5.o 


462 


464 


474 


483 


492 


502 


5" 


521 


53° 


539 


549 


6 
7 
8 


6.0 


463 


558 


567 


577 


586 


596 


605 


614 


624 


6 33 


642 


7.0 

8.0 


464 


652 


661 


671 


680 


689 


699 


708 


717 


727 


73 6 


9 


9.0 


465 


745 


755 


764 


773 


783 


792 


801 


811 


820 


829 




466 


839 


848 


857 


867 


876 


885 


894 


904 


9*3 


922 




467 


93 2 


941 


95° 


960 


969 


978 


987 


997 


*oo6 


*oi5 




468 


67 025 


°34 


043 


052 


062 


071 


080 


089 


099 


108 




469 


117 


127 


136 


i45 


154 


164 


173 


182 


191 


201 




470 


210 


219 


228 


2 37 


247 


256 


265 


274 


284 


2 93 




47i 


302 


311 


321 


33° 


339 


348 


357 


3 6 7 


376 


385 




9 

0.0 


472 


394 


403 


413 


422 


43i 


440 


449 


459 


468 


477 


1 


473 


486 


495 


5°4 


5 J 4 


5 2 3 


r ~> 

5o- 


54i 


55° 


560 


5 6 9 


2 


1. 


8 


474 


578 


587 


596 


605 


614 


624 


633. 


642 


651 


660 


3 


2. 

3. 


7 
6 


475 


669 


679 


688 


697 


706 


715 


724 


733 


742 


752 


5 


4- 


5 


476 


761 


770 


779 


788 


797 


806 


8i5 


825 


834 


843 


6 
7 
8 


5- 

6 


4 


477 


852 


861 


870 


879 


888 


897 


906 


916 


925 


934 


7- 



2 


478 


943 


952 


961 


970 


979 


988 


997 


*oo6 


*oi5 


*024 


9 


8.1 


479 


68 034 


043 


052 


061 


070 


O79 


088 


097 


106 


115 




480 


124 


*33 


142 


I5 1 


160 


169 


178 


187 


196 


205 




481 


215 


224 


233 


242 


25 1 


260 


269 


278 


287 


296 




482 


3°5 


3*4 


323 


33 2 


34i 


35° 


359 


368 


377 


386 




483 


395 


404 


4i3 


422 


43i 


440 


449 


458 


467 


476 




484 


485 


494 


502 


5ii 


520 


529 


538 


547 


556 


565 




485 


574 


583 


592 


601 


610 


619 


628 


637 


646 


655 




g 


486 


664 


673 


681 


690 


699 


708 


717 


726 


735 


744 


1 


0.8 


487 


753 


762 


771 


780 


789 


797 


806 


815 


824 


833 


2 


1 


6 


488 


842 


851 


860 


869 


878 


886 


895 


904 


9i3 


922 


3 

4 


2 

3 


4 
.2 


489 


93i 


940 


949 


958 


966 


975 


984 


993 


*002 


*OII 


5 
6 

7 


4 




8 

.6 


490 


69 020 


028 


o37 


046 


°55 


064 


°73 


082 


O9O 


099 


4 
5 


491 


108 


117 


126 


135 


144 


152 


161 


170 


179 


188 


8 


6 


• 4 


492 


197 


205 


214 


223 


232 


241 


249 


258 


267 


276 


9 


7.2 


493 


285 


294 


302 


3 11 


320 


329 


338 


346 


355 


3 6 4 




494 


373 


381 


39° 


399 


408 


4i7 


425 


434 


443 


452 




495 


461 


469 


478 


487 


496 


5°4 


5i3 


522 


53i 


539 




496 


548 


557 


566 


574 


583 


592 


601 


609 


618 


627 




497 


636 


644 


653 


662 


671 


679 


688 


697 


7°5 


714 




498 


7 2 3 


732 


740 


749 


758 


767 


775 


784 


793 


801 




499 


810 


819 


827 


836 


845 


854 


862 


871 


880 


888 





IO3 



LOGARITHMS OF NUMBERS. 



No. 





I 


2 


3 


4 


5 


6 


7 


8 


9 


Pp. Pts. 


500 


69 897 


906 


914 


9 2 3 


932 


940 


949 


958 


966 


975 




50I 


984 


992 = 


^001 : 


"OTO ' 


1=018 *o27 - 


^036 '- 


k 044 : 


•o53 : 


^062 




502 


70 070 


079 


088 


O96 


io5 


114 


122 


131 


140 


148 




503 


157 


165 


174 


183 


191 


200 


209 


217 


226 


234 




504 


243 


252 


260 


269 


278 


286 


295 


3°3 


312 


321 




505 


3 2 9 


338 


346 


355 


3 6 4 


37 2 


381 


389 


398 


406 




506 


■ 4i5 


424 


43 2 


441 


449 


458 


467 


475 


484 


49 2 




507 


5°i 


5°9 


5i8 


526 


535 


544 


552 


56i 


5 6 9 


578 




508 


586 


595 


603 


612 


621 


629 


638 


646 


655 


663 


I 


9 

3. 9 


509 


672 


680 


689 


697 


706 


714 


723 


73i 


740 


749 


2 


1.8 


5IO 


757 


766 


774 


783 


791 


800 


808 


817 


825 


834 


3 


2.7 ! 
1 f\ 


511 


842 


851 


859 


868 


876 


885 


893 


902 


910 


919 


4 
5 


3- ° 
4-5 


512 


927 


935 


944 


95 2 


961 


969 


978 


986 


995 


*oc>3 


6 

7 
8 


5-4 
6 x 


513 


71 012 


020 


029 


037 


046 


o54 


063 


071 


079 


088 


u ■ ) 
7.2 


514 


096 


i°5 


113 


122 


130 


i39 


147 


155 


164 


172 


9 


8.1 \ 


515 


181 


189 


198 


206 


214 


223 


231 


240 


248 


2 57 




5l6 


265 


273 


282 


290 


299 


3°7 


315 


3 2 4 


33 2 


34i 




517 


349 


357 


366 


374 


383 


39 1 


399 


408 


416 


425 




5l8 


433 


441 


45° 


458 


466 


475 


483 


492 


500 


508 




519 


5i7 


525 


533 


542 


55o 


559 


567 


575 


584 


59 2 




520 


600 


609 


617 


625 


634 


642 


650 


659 


667 


675 




521 


684 


692 


700 


709 


717 


7 2 5 


734 


742 


75° 


759 




a 


522 


767 


775 


784 


792 


800 


809 


817 


825 


834 


842 


1 




0.8 


523 


850 


858 


867 


875 


883 


892 


900 


908 


917 


9 2 5 


2 


1.6 


524 


933 


941 


95° 


958 


966 


975 


9S3 


991 


999 


*oo8 


3 

4 


2.4 
3-2 


525 


72 016 


024 


032 


041 


049 


o57 


066 


074 


082 


090 


5 


4.0 


526 


099 


107 


ii5 


123 


132 


140 


148 


156 


165 


173 


6 

7 


4.8 
5.6 


527 


181 


189 


198 


206 


214 


222 


230 


2 39 


247 


255 


8 


6.4 


528 


263 


272 


280 


288 


296 


3°4 


3*3 


321 


3 2 9 


337 


9 


7-2 


529 


346 


354 


362 


37° 


378 


387 


395 


403 


411 


419 




530 


428 


43 6 


444 


452 


460 


469 


477 


485 


493 


5°i 




531 


5°9 


5i8 


526 


534 


542 


55° 


55S 


5 6 7 


575 


583 




532 


59i 


599 


607 


616 


624 


632 


640 


648 


656 


665 




533 


673 


681 


689 


697 


7°5 


713 


722 


73° 


738 


746 




534 


754 


762 


770 


779 


787 


795 


803 


811 


819 


827 




535 


835 


843 


852 


860 


868 


876 


884 


892 


900 


908 




7 

0.7 


536 


916 


9 2 5 


933 


941 


949 


957 


965 


973 


981 


989 


1 


537 


997 


*oo6 


*oi4 


*02 2 


*030 


*c>38 


+046 


*o54 


*o62 


+070 


2 


i-4 


538 


73 078 


086 


094 


I02 


in 


119 


127 


135 


143 


151 


3 

4 


2 . 1 
2.8 


539 


159 


167 


175 


183 


191 


199 


207 


2 *5 


223 


231 


5 
6 

7 


3-5 


540 


239 


247 


255 


263 


272 


280 


288 


296 


3°4 


312 


4. 2 
4.9 


54i 


320 


328 


33 6 


344 


35 2 


360 


368 


37 6 


384 


39 2 


8 


5-6 


542 


400 


408 


416 


424 


43 2 


440 


448 


45 6 


464 


472 


9 


6-3 


543 


480 


488 


496 


5°4 


5i 2 


520 


528 


53 6 


544 


55 2 




544 


560 


568 


576 


584 


59 2 


600 


608 


616 


624 


632 




545 


640 


648 


656 


664 


672 


679 


687 


695 


7°3 


711 




546 


719 


727 


735 


743 


75i 


759 


767 


775 


783 


791 




547 


799 


807 


815 


823 


830 


838 


846 


854 


862 


870 




548 


878 


886 


894 


902 


910 


918 


926 


933 


941 


949 




549 


957 


965 


973 


981 


989 


997 


*oo5 


*oi3 


*020 


*028 




1 


74 























104 



LOGARITHMS OF NUMBERS. 



No. 





1 


2 


3 


4 


5 


6 


7 


8 


9 


Pp.Pts. 


550 


74 036 


044 


052 


060 


068 


076 


084 


092 


099 


107 




551 


115 


123 


13 1 


139 


147 


155 


162 


170 


178 


186 




552 


194 


202 


210 


218 


225 


233 


241 


249 


257 


265 




553 


273 


280 


288 


296 


3°4 


312 


320 


327 


335 


343 




554 


35i 


359 


3 6 7 


374 


382 


39° 


398 


406 


414 


421 




555 


429 


437 


445 


453 


461 


468 


476 


484 


492 


500 




556 


5°7 


5i5 


5 2 3 


53i 


539 


547 


554 


562 


57° 


578 




557 


586 


593 


601 


609 


617 


624 


632 


640 


648 


656 




558 


663 


671 


679 


687 


695 


702 


710 


718 


726 


733 




559 


741 


749 


757 


764 


772 


780 


788 


796 


803 


811 




560 


819 


827 


834 


842 


850 


858 


865 


873 


881 


889 




561 


896 


904 


912 


920 


927 


935 


943 


95° 


958 


966 




562 


974 


981 


989 


997 


*oc>5 


*OI2 


*020 


*028 


*o35 


*°43 




g 


563 


75 °Si 


°59 


066 


074 


082 


089 


097 


105 


113 


120 


I 


8 


564 


128 


136 


143 


151 


i59 


166 


174 


182 


189 


197 


2 


1 


6 


565 


205 


213 


220 


228 


236 


243 


251 


259 


266 


274 


3 

4 


2 
3 


4 
2 


566 


282 


289 


297 


3°5 


312 


32O 


328 


335 


343 


35i 


5 


4 





567 


358 


366 


374 


381 


389 


397 


404 


412 


420 


427 


6 

7 
8 


4 


8 
6 


568 


435 


442 


45° 


458 


465 


473 


481 


488 


496 


5°4 


5 
6 


4 


569 


5ii 


5i9 


526 


534 


542 


549 


557 


565 


572 


580 


9 


7 


2 


57o 


587 


595 


603 


610 


618 


626 


6 33 


641 


648 


656 




57i 


664 


671 


679 


686 


694 


702 


709 


717 


724 


732 




572 


740 


747 


755 


762 


770 


778 


785 


793 


800 


808 




573 


815 


823 


831 


838 


846 


853 


861 


868 


876 


884 




574 


891 


899 


906 


914 


921 


929 


937 


944 


95 2 


959 




575 


967 


974 


982 


989 


997 


*oo5 


*OI2 


*020 


*02 7 


*Q35 




576 


76 042 


050 


o57 


065 


072 


080 


087 


°95 


103 


no 




577 


118 


125 


133 


140 


148 


i55 


163 


170 


178 


185 




578 


193 


200 


208 


215 


223 


230 


238 


245 


253 


260 




579 


268 


275 


283 


290 


298 


3°5 


313 


320 


328 


335 




580 


343 


35° 


358 


365 


373 


380 


388 


395 


403 


410 






581 


418 


425 


433 


440 


448 


455 


462 


470 


477 


485 


1 


7 
0. 7 


582 


492 


500 


5°7 


5i5 


522 


53° 


537 


545 


55 2 


559 


2 


1 


4 


583 


567 


574 


582 


589 


597 


604 


612 


619 


626 


634 


3 

4 
5 


2 

2 


1 
8 


584 


641 


649 


656 


664 


671 


678 


686 


693 


701 


708 


3 


S 


585 


716 


7 2 3 


73° 


738 


745 


753 


760 


768 


775 


782 


6 


4 


2 


586 


790 


797 


805 


812 


819 


827 


834 


842 


849 


856 


7 
8 


4 

5 


9 
6 


587 


864 


871 


879 


886 


893 


901 


908 


916 


9 2 3 


93° 


9 


6 


3 


588 


938 


945 


953 


960 


967 


975 


982 


989 


997 


*oo4 




589 


77 012 


019 


026 


034 


041 


048 


056 


063 


070 


078 




590 


085 


°93 


100 


107 


115 


122 


129 


137 


144 


.^ 




59i 


159 


166 


173 


181 


188 


i95 


203 


210 


217 


225 




592 


232 


240 


247 


254 


262 


269 


276 


283 


291 


298 




593 


3°5 


3*3 


320 


3 2 7 


335 


342 


349 


357 


3 6 4 


37 1 




594 


379 


386 


393 


401 


408 


4i5 


422 


43° 


437 


444 




595 


45 2 


459 


466 


474 


481 


488 


495 


5°3 


5i° 


517 




596 


525 


53 2 


539 


546 


554 


561 


568 


576 


583 


590 




597 


597 


605 


612 


619 


627 


634 


641 


648 


656 


663 




598 


670 


677 


685 


692 


699 


706 


714 


721 


728 


735 




599 


743 


75° 


J 757 


764 


772 


779 


786 


793 


801 


808 





I°5 



LOGARITHMS OF NUMBERS. 



No. 


O 


1 


2 


3 


4 


5 


6 


7 


8 


9 


Pp. Pts. 


600 


77 815 


822 


830 


837 


844 


851 


859 


866 


873 


880 






6oi 


887 


895 


902 


90Q 


916 


924 


93i 


938 


945 


952 






602 


960 


967 


974 


981 


988 


996 


*oo3 


*OIO 


*oi7 


*025 






603 


78 032 


o39 


046 


°53 


061 


068 


°75 


082 


089 


097 






604 


104 


in 


118 


i 2 5 


132 


140 


147 


154 


161 


168 






605 


176 


183 


190 


197 


204 


211 


219 


226 


233 


240 






606 


247 


254 


262 


269 


276 


283 


290 


297 


3°5 


312 






607 


3i9 


326 


333 


34o 


347 


355 


362 


369 


376 


383 






608 


39° 


398 


405 


412 


419 


426 


433 


440 


447 


455 


i 

I 


8 


609 


462 


469 


476 


483 


490 


497 


5°4 


512 


5i9 


526 


2 I 


6 


6lO 


533 


54o 


547 


554 


56i 


569 


576 


583 


59° 


597 


3 2 


4 


6ll 


604 


611 


618 


625 


6 33 


640 


647 


654 


661 


668 


4 3 

5 4 


2 




6l2 


675 


682 


689 


696 


704 


711 


718 


725 


732 


739 


6 4 


8 


613 


746 


753 


760 


767 


774 


781 


789 


796 


803 


810 


7 5 

8 6 


6 
4 


614 


817 


824 


831 


838 


845 


852 


859 


866 


873 


880 


9 7 


2 


615 


888 


895 


902 


909 


916 


923 


93° 


937 


944 


95i 






616 


958 


9 6 5 


972 


9^9 


986 


993 


*ooo 


*oo7 


*oi4 


*02I 






617 


79 029 


036 


043 


050 


°57 


064 


071 


078 


085 


O92 






618 


099 


106 


"3 


120 


127 


134 


141 


148 


155 


l62 






619 


169 


176 


183 


190 


197 


204 


211 


218 


225 


232 






620 


239 


246 


2 53 


260 


' 267 


274 


28l 


288 


295 


302 






621 


3°9 


316 


323 


33° 


337 


344 


351 


358 


365 


372 






622 


379 


386 


393 


400 


407 


414 


421 


428 


435 


442 


1 


7 

7 


623 


449 


45 6 


463 


470 


477 


484 


491 


498 


5°5 


5" 


2 1 


4 


024 


5i8 


525 


53 2 


539 


546 


553 


560 


567 


574 


58l 


3 2 

A. 2 


1 
8 


625 


588 


595 


602 


669 


616 


623 


630 


637 


644 


65O 


5 3 


S 


626 


6 57 


664 


671 


678 


685 


692 


699 


706 


7 J 3 


72O 


6 4 


2 


627 


727 


734 


741 


748 


754 


761 


768 


775 


782 


789 


7 4 

8 5 


9 

6 


628 


796 


803 


810 


817 


824 


83* 


837 


844 


851 


858 


9 6 


3 


629 


865 


872 


879 


886 


893 


900 


906 


9i3 


920 


927 






630 


934 


941 


948 


955 


962 


969 


975 


982 


989 


996 






631 


80 003 


010 


017 


024 


030 


°37 


044 


051 


058 


065 






632 


072 


079 


085 


092 


099 


106 


113 


120 


127 


134 






633 


140 


147 


i54 


161 


168 


i75 


182 


188 


195 


202 






634 


209 


216 


223 


229 


236 


243 


250 


257 


264 


271 






635 


277 


284 


291 


298 


3°5 


312 


318 


325 


332 


339 


( 




636 


346 


353 


359 


366 


373 


380 


387 


393 


400 


407 


1 


6 


637 


414 


421 


428 


434 


441 


448 


455 


462 


468 


475 


2 1 


2 


638 


482 


489 


496 


502 


5°9 


5i6 


523 


53° 


536 


543 


3 1 

4 2 


8 
4 


639 


55° 


557 


564 


57° 


577 


S84 


59i 


598 


604 


611 


5 3 





640 


618 


625 


632 


638 


645 


652 


659 


665 


672 


679 


6 3 

7 4 


6 
2 


641 


686 


6 93 


699 


706 


7 J 3 


720 


726 


733 


740 


747 


8 4 


8 


642 


754 


760 


767 


774 


781 


787 


794 


801 


808 


814 


9 5 


4 


643 


821 


828 


835 


841 


848 


855 


862 


868 


875 


882 






644 


889 


895 


902 


909 


916 


922 


929 


93 6 


943 


949 






645 


956 


963 


969 


976 


983 


990 


996 


*oo3 


*OIO 


*oi7 






646 


81 023 


030 


°37 


043 


050 


°57 


064 


070 


077 


084 






647 


090 


097 


104 


in 


117 


124 


I3 1 


137 


144 


151 






648 


158 


164 


171 


178 


184 


191 


198 


204 


211 


218 






649 


224 


231 


238 


245 


251 


258 


265 


271 


278 


28s 







106 



LOGARITHMS OF NUMBERS. 



No. 





1 


2 


3 


4 


5 


6 


7 


8 


9 


Pp. Pts. 


650 


81 291 


298 


3°5 


3ii 


318 


325 


33* 


338 


345 


35i 




651 


353 


365 


37i 


378 


385 


39i 


398 


405 


411 


418 




652 


425 


43i 


438 


445 


45i 


458 


465 


471 


478 


485 




653 


491 


498 


5°5 


5ii 


5i8 


525 


53i 


538 


544 


55i 




654 


558 


5 6 4 


57i 


578 


584 


59i 


598 


604 


611 


617 




655 


624 


631 


6 37 


644 


651 


657 


664 


671 


677 


684 




656 


690 


697 


704 


710 


717 


723 


73° 


737 


743 


75° 




657 


757 


763 


770 


776 


783 


790 


796 


803 


809 


816 




658 


823 


829 


836 


842 


849 


856 


862 


869 


875 


882 




659 


889 


895 


902 


908 


9i5 


921 


928 


935 


941 


948 




660 


954 


961 


968 


974 


981 


987 


994 


*ooo 


*oo7 


*oi4 




66l 


82 020 


027 


°33 


040 


046 


°53 


060 


066 


°73 


079 




662 


086 


092 


099 


105 


112 


119 


125 


132 


138 


145 






663 


151 


158 


164 


171 


178 


184 


191 


197 


204 


210 


1 


7 
0.1 


664 


217 


223 


230 


236 


243 


249 


256 


263 


269 


276 


2 


1 


4 


665 


282 


289 


295 


302 


308 


3*5 


321 


328 


334 


34i 


3 

4 


2 
2 


1 
8 


666 


347 


354 


360 


3C7 


373 


380 


387 


393 


400 


406 


5 


3 


5 


667 


413 


419 


426 


A3 2 


439 


445 


452 


458 


465 


471 


6 

7 
8 


4 


2 


668 


478 


484 


491 


497 


5°4 


5io 


5i7 


523 


53° 


536 


4 
5 


9 
6 


669 


543 


549 


556 


562 


569 


575 


582 


588 


595 


601 


9 


6 


3 


670 


607 


614 


620 


627 


6 33 


640 


646 


6 53 


659 


666 




671 


672 


679 


.685 


692 


698 


705 


711 


718 


724 


73° 




672 


737 


743 


75° 


75 6 


763 


769 


776 


782 


789 


795 




673 


802 


808 


814 


821 


827 


834 


840 


847 


853 


860 




674 


866 


872 


879 


885 


892 


898 


9°5 


911 


918 


924 




675 


930 


937 


943 


95° 


956 


963 


969 


975 


982 


988 




676 


995 


*OOI 


*oo8 


*oi4 


*020 


^027 


*°33 


*04o 


#046 


^052 




677 


8 3 °59 


065 


072 


078 


08 5 


091 


097 


104 


no 


117 




678 


123 


129 


136 


142 


149 


155 


161 


168 


174 


181 




679 


187 


193 


200 


206 


213 


219 


225 


232 


238 


245 




680 


251 


257 


264 


270 


276 


283 


289 


296 


302 


308 




6 

O-fi 


681 


3i5 


321 


327 


334 


340 


347 


353 


359 


366 


37 2 


1 


682 


37* 


385 


39i 


398 


404 


410 


417 


423 


429 


43 6 


2 


1 


2 


683 


442 


448 


455 


461 


467 


474 


480 


487 


493 


499 


3 

4 


1 

2 


8 
4 


684 


506 


512 


5i8 


525 


531 


537 


544 


55o 


556 


5 6 3 


5 


3 





685 


5 6 9 


575 


582 


588 


594 


601 


607 


613 


620 


626 


6 

f7 


3 


6 


68*6 


632 


639 


645 


651 


658 


664 


670 


677 


683 


689 


7 
8 


4 
4 


8 


687 


696 


702 


708 


715 


721 


727 


734 


740 


746 


753 


9 


5 


•4 


688 


759 


7 6 5 


771 


778 


784 


790 


797 


803 


809 


816 




689 


822 


828 


835 


841 


847 


853 


860 


866 


872 


879 




690 


885 


891 


897 


904 


910 


916 


923 


929 


935 


942 




691 


948 


954 


960 


967 


973 


979 


985 


992 


998 


*oo4 




692 


84 on 


017 


023 


029 


036 


042 


048 


o55 


061 


067 




693 


°73 


080 


086 


092 


098 


105 


in 


117 


123 


130 




694 


136 


142 


148 


155 


161 


167 


i73 


180 


186 


192 




695 


198 


205 


211 


217 


223 


230 


236 


242 


248 


255 




696 


261 


267 


273 


280 


286 


292 


298 


3°5 


311 


3 J 7 




697 


3 2 3 


33° 


33 6 


342 


348 


354 


361 


3 6 7 


373 


379 




698 


386 


392 


398 


404 


410 


417 


423 


429 


435 


442 




699 


448 


454 


460 


466 


473 


479 


485 


491 


497 


5°4 





107 



LOGARITHMS OF NUMBERS. 



No. 





I 


2 


3 


4 


5 


6 


7 


8 


9 


Pp. Pts. 


700 


84 510 


5i6 


522 


528 


535 


54i 


547 


553 


559 


566 






701 


572 


578 


584 


590 


597 


603 


609 


615 


621 


628 






702 


634 


640 


646 


652 


658 


665 


671 


677 


683 


689 






703 


696 


702 


708 


714 


720 


726 


733 


739 


745 


75i 






704 


757 


763 


770 


776 


782 


788 


794 


800 


807 


813 






705 


819 


825 


831 


837 


844 


850 


856 


862 


868 


874 






706 


880 


887 


893 


899 


9°5 


911 


917 


924 


93° 


93 6 






707 


942 


948 


954 


960 


967 


973 


979 


985 


991 


997 






708 


85 003 


009 


016 


022 


028 


o34 


040 


046 


052 


058 


1 


7 
7 
4 


709 


065 


071 


077 


083 


089 


°95 


IOI 


107 


114 


120 


2 1 


710 


126 


132 


138 


144 


i5° 


156 


163 


169 


175 


181 


3 2 


1 
8 
5 


711 


187 


193 


199 


205 


211 


217 


224 


230 


236 


242 


4 2 

5 3 


712 


248 


254 


260 


266 


272 


278 


285 


291 


297 


3°3 


6 4 


2 


713 


3°9 


3i5 


321 


327 


333 


339 


345 


352 


358 


3 6 4 


7 4 

8 5 


9 
6 


714 


37° 


376 


382 


388 


394 


400 


406 


412 


418 


425 


9 6 


3 


715 


43i 


437 


443 


449 


455 


461 


467 


473 


479 


485 






716 


491 


497 


5°3 


5°9 


5i6 


522 


528 


534 


54o 


546 






717 


55 2 


558 


5 6 4 


57° 


576 


582 


588 


594 


600 


606 






718 


612 


618 


625 


631 


637 


643 


649 


655 


661 


667 






719 


673 


679 


685 


691 


697 


7°3 


709 


7i5 


721 


727 






720 


733 


739 


745 


75i 


757 


7 6 3 


769 


775 


781 


788 






721 


794 


800 


806 


812 


818 


824 


830 


836 


842 


848 




6 


722 


854 


860 


866 


872 


878 


884 


890 


896 


902 


908 


( 
I 


723 


914 


920 


926 


932 


938 


944 


95° 


956 


962 


968 


2 1 


2 


724 


974 


980 


986 


992 


998 


*oo4 


*OIO 


*oi6 


*022 


*028 


3 1 

4 2 

5 3 


8 


725 


86 034 


040 


046 


052 


058 


064 


070 


076 


082 


088 


O 


726 


094 


100 


106 


112 


118 


124 


130 


136 


141 


147 


6 3 

7 4 

8 4 


6 


727 


153 


159 


165 


171 


177 


183 


189 


195 


20I 


207 


8 


728 


213 


219 


225 


231 


2 37 


243 


249 


255 


26l 


267 


9 S 


4 


729 


273 


279 


285 


291 


297 


3°3 


308 


3i4 


32O 


326 






730 


33 2 


338 


344 


35° 


356 


362 


368 


374 


38o 


386 






731 


392 


398 


404 


410 


4i5 


421 


427 


433 


439 


445 






732 


45i 


457 


463 


469 


475 


481 


487 


493 


499 


5°4 






733 


5io 


5i6 


522 


528 


534 


54o 


546 


552 


558 


5 6 4 






734 


57° 


576 


58i 


587 


593 


599 


605 


611 


617 


623 






735 


629 


635 


641 


646 


652 


658 


664 


670 


676 


682 






736 


688 


694 


700 


7o5 


711 


717 


723 


729 


735 


741 


1 


5 

5 


737 


747 


753 


759 


764 


770 


776 


782 


788 


794 


800 


2 1 





738 


806 


812 


817 


823 


829 


835 


841 


847 


853 


859 


3 1 

4 2 


5 



739 


864 


870 


876 


882 


888 


894 


900 


906 


911 


917 


5 2 


5 


740 


9 2 3 


929 


935 


941 


947 


953 


958 


964 


970 


976 


6 3 

*7 3 




r 


74i 


982 


988 


994 


999 


*oo5 


*OII 


*oi7 


*023 


*029 


*°35 


8 4 


O 


742 


87 040 


046 


052 


058 


064 


070 


o75 


081 


087 


°93 


9 4 


5 


743 


099 


105 


in 


116 


122 


128 


134 


140 


146 


151 






744 


157 


163 


169 


175 


181 


186 


192 


198 


204 


210 






745 


216 


221 


227 


233 


239 


245 


251 


256 


262 


268 






746 


274 


280 


286 


291 


297 


3°3 


3°9 


315 


320 


326 






747 


332 


338 


344 


349 


355 


361 


367 


373 


379 


384 






748 


39° 


39 6 


402 


408 


413 


419 


425 


43i 


437 


442 






749 


448 


454 


460 


466 


47i 


477 


483 


489 


495 


500 







108 



LOGARITHMS OF NUMBERS. 



No. 





1 


2 


3 


4 


5 


6 


7 


8 


9 


Pp.Pts. 


750 


87 506 


512 


5iS 


523 


529 


535 


54i 


547 


552 


558 




751 


5 6 4 


57° 


576 


58i 


587 


593 


599 


604 


610 


616 




752 


622 


628 


6 33 


6 39 


645 


651 


656 


662 


668 


674 




753 


679 


685 


691 


697 


7°3 


708 


714 


720 


726 


73i 




754 


737 


743 


749 


754 


760 


766 


772 


777 


783 


789 




755 


795 


800 


806 


812 


818 


823 


829 


835 


841 


846 




756 


852 


858 


864 


869 


875 


881 


887 


892 


898 


904 




757 


910 


9i5 


921 


927 


933 


938 


944 


95° 


955 


961 




758 


967 


973 


978 


984 


990 


996 


*OOI 


*oo7 


*°i3 


*oi8 




759 


88 024 


030 


036 


041 


047 


°53 


058 


064 


070 


076 




760 


081 


087 


°93 


098 


104 


no 


116 


121 


127 


133 




761 


138 


144 


150 


*5 6 


161 


167 


173 


178 


184 


190 




762 


195 


201 


207 


213 


218 


224 


230 


235 


241 


247 




763 


252 


258 


264 


270 


275 


281 


287 


292 


298 


3°4 


1 


u 
6 


764 


3°9 


3i5 


321 


326 


332 


338 


343 


349 


355 


360 


2 


1 


2 


765 


366 


372 


377 


383 


389 


395 


400 


406 


412 


4i7 


3 

4 


1 
2 


8 


766 


423 


429 


434 


440 


446 


45i 


457 


463 


46S 


474 


5 


3 


O 


767 


480 


485 


491 


497 


502 


508 


513 


5i9 


525 


53° 


6 

7 
8 


3 


6 


768 


536 


542 


547 


553 


559 


5 6 4 


57o 


576 


58i 


587 


4 
4 


8 


769 


593 


598 


604 


610 


615 


621 


627 


632 


638 


643 


9 


5 


4 


770 


649 


655 


660 


666 


672 


677 


683 


689 


694 


700 




771 


7°5 


711 


717 


722 


728 


734 


739 


745 


75o 


756 




772 


762 


767 


773 


779 


784 


790 


795 


801 


807 


812 




773 


818 


824 


829 


835 


840 


846 


852 


857 


863 


868 




774 


874 


880 


885 


891 


897 


902 


908 


9i3 


919 


925 




775 


93° 


93 6 


941 


947 


953 


958 


964 


969 


975 


981 




776 


986 


992 


997 


*oo3 


*oo9 


*oi4 


*020 


*025 


*o3i 


*<>37 




777 


89 042 


048 


°53 


°59 


064 


070 


O76 


081 


087 


092 




778 


098 


104 


109 


"5 


120 


126 


131 


137 


143 


148 




779 


154 


i59 


165 


170 


176 


182 


187 


193 


198 


204 




780 


209 


215 


221 


226 


232 


237 


243 


248 


254 


260 




781 


265 


271 


276 


282 


287 


293 


298 


3°4 


310 


3i5 


1 


5 

c 


782 


321 


326 


332 


337 


343 


348 


354 


360 


365 


3n 


2 


1 





783 


376 


382 


387 


393 


398 


404 


409 


4i5 


421 


426 


3 

4 


1 
2 


5 



784 


43 2 


437 


443 


448 


454 


459 


465 


470 


476 


481 


5 


2 


5 


785 


487 


492 


498 


5°4 


5°9 


5i5 


520 


526 


53i 


537 


6 


3 





786 


542 


548 


553 


559 


5 6 4 


57° 


575 


58i, 


586 


592 


7 
8 


3 

4 


5 



787 


597 


603 


609 


614 


620 


625 


631 


636 


642 


647 


9 


4 


5 


788 


653 


658 


664 


669 


675 


680 


686 


691 


697 


702 




789 


708 


7i3 


719 


724 


73° 


735 


741 


746 


752 


757 




790 


7 6 3 


768 


774 


779 


785 


790 


796 


801 


807 


812 




791 


818 


823 


829 


834 


840 


845 


851 


856 


862 


867 




792 


873 


878 


883 


889 


894 


900 


9°5 


911 


916 


922 




793 


927 


933 


938 


944 


949 


955 


960 


966 


971 


977 




794 


982 


988 


993 


998 


*oo4 


*oo9 


*oi5 


*020 


*026 


*°3 T 




795 


90 037 


042 


048 


°53 


o59 


064 


069 


°75 


080 


086 




796 


091 


097 


102 


108 


ii3 


119 


124 


129 


135 


140 




797 


146 


151 


157 


162 


168 


173 


179 


184 


189 


195 




798 


200 


206 


211 


217 


222 


227 


233 


238 


244 


249 




799 


255 


260 


266 


271 


276 


282 


287 


293 


298 


3°4 





109 



LOGARITHMS OF NUMBERS. 



No. 





I 
3i4 


2 

320 


3 


4 


5 


6 

342 


7 


8 


9 


Pp. Pts. 


800 


90 309 


325 


33i 


33 6 


347 


352 


358 




8oi 


3 6 3 


3 6 9 


374 


380 


385 


39° 


396 


401 


407 


412 




802 


417 


423 


428 


434 


439 


445 


45° 


455 


461 


466 




803 


472 


477 


482 


488 


493 


499 


5°4 


5°9 


5i5 


520 




804 


526 


53i 


53 6 


542 


547 


553 


558 


563 


5 6 9 


574 




805 


580 


585 


59° 


596 


601 


607 


612 


617 


623 


628 




806 


634 


6 39 


644 


650 


655 


660 


666 


671 


677 


682 




807 


687 


693 


698 


7°3 


709 


714 


720 


725 


73° 


736 




808 


741 


747 


752 


757 


7 6 3 


768 


773 


779 


784 


789 




809 


795 


800 


806 


8n 


816 


822 


827 


832 


838 


843 




8lO 


849 


854 


859 


865 


870 


875 


881 


886 


891 


897 




8ll 


902 


907 


9i3 


918 


924 


929 


934 


940 


945 


95° 




8l2 


95 6 


961 


966 


972 


977 


982 


988 


993 


998 


*oo4 




5 


813 


91 009 


014 


020 


025 


030 


036 


041 


046 


052 


°57 


I 


0.6 


814 


062 


068 


°73 


078 


084 


089 


094 


100 


io 5 


no 


2 


1 


2 


815 


116 


121 


126 


132 


t-37 


142 


148 


153 


158 


164 


3 

4 


1 
2 


8 
4 


816 


169 


174 


180 


185 


190 


196 


201 


206 


212 


217 


5 


3 





817 


222 


228 


2 33 


238 


243 


249 


254 


2 59 


265 


270 


6 

•7 


3 

4 


6 
2 


818 


275 


2S1 


286 


291 


297 


302 


3°7 


312 


318 


323 


8 


4 


8 


819 


328 


334 


339 


344 


35° 


355 


360 


3 6 5 


37 1 


37 6 


9 


5-4 


82O 


381 


387 


392 


397 


403 


408 


4i3 


418 


424 


429 




821 


434 


440 


445 


45° 


455 


461 


466 


471 


477 


482 




822 


487 


492 


498 


5°3 


508 


5i4 


5i9 


524 


529 


535 




823 


54o 


545 


55i 


556 


561 


566 


572 


577 


582 


587 




824 


593 


598 


603 


609 


614 


619 


624 


630 


635 


640 




825 


645 


651 


656 


661 


666 


672 


677 


682 


687 


6 93 




826 


698 


7°3 


709 


714 


719 


724 


73° 


735 


740 


745 




827 


75i 


756 


761 


766 


772 


777 


782 


787 


793 


798 




828 


803 


8c8 


814 


819 


824 


829 


834 


840 


845 


850 




829 


855 


861 


866 


871 


876 


882 


887 


892 


897 


9°3 




83O 


908 


9 r 3 


91S 


924 


929 


934 


939 


944 


95° 


955 






831 


960 


9 6 5 


971 


976 


981 


986 


991 


997 


*002 


*oo7 


1 


5 
0. <; 


832 


92 012 


018 


023 


028 


°33 


038 


044 


049 


054 


o59 


2 


1 





833 


065 


070 


o75 


080 


085 


091 


096 


IOI 


I06 


in 


3 

4 


1 
2 


5 



834 


117 


122 


127 


132 


137 


143 


148 


153 


158 


163 


5 


2 


5 


835 


169 


174 


179 


184 


189 


195 


200 


205 


2IO 


215 


6 

17 


3 




5 



836 


221 


226 


231 


236 


241 


247 


252 


257 


262 


267 


/ 
8 


3 

4 


837 


273 


278 


283 


288 


293 


298 


3°4 


3°9 


3 J 4 


3i9 


9 


4 


5 


838 


3 2 4 


33° 


335 


34o 


345 


35° 


355 


361 


366 


37i 




839 


376 


381 


387 


392 


397 


402 


407 


412 


418 


423 




840 


428 


433 


438 


443 


449 


454 


459 


464 


469 


474 




841 


480 


485 


490 


495 


500 


505 


5ii 


5i6 


521 


526 




842 


53i 


536 


542 


54-7 


552 


557 


562 


567 


572 


578 




843 


583 


588 


593 


598 


603 


609 


614 


619 


624 


629 




844 


634 


6 39 


645 


650 


655 


660 


665 


670 


675 


681 




845 


686 


691 


696 


701 


706 


711 


716 


722 


727 


732 




846 


737 


742 


747 


75 2 


•758 


7 6 3 


768 


773 


778 


783 




847 


788 


793 


799 


804 


809 


814 


819 


824 


829 


834 




848 


840 


845 


850 


855 


860 


865 


870 


875 


881 


886 




849 


891 


896 


901 


906 


911 


916 


921 


927 


932 


937 





no 



LOGARITHMS OF NUMBERS. 



No. 





1 


2 


3 


4 


5 


6 


7 


8 


9 


1 ■ 

Pp. Pts. 


850 


92 942 


947 


952 


957 


962 


967 


973 


978 


983 


988 




851 


993 


998 


*oo3 


*oo8 


*oi3 


*oi8 


*024 


*029 


*c>34 


*°39 




852 


93 °44 


049 


o54 


°59 


064 


069 


°75 


080 


085 


090 




853 


o95 


100 


105 


no 


ii5 


120 


125 


131 


136 


141 




854 


146 


151 


156 


161 


166 


171 


176 


181 


186 


192 




855 


197 


202 


207 


212 


217 


222 


227 


232 


237 


242 




856 


247 


252 


258 


263 


268 


273 


278 


283 


288 


293 




857 


298 


3°3 


308 


3*3 


318 


323 


328 


334 


339 


344 




858 


349 


354 


359 


3 6 4 


3 6 9 


374 


379 


384 


389 


394 


I 


6 

* 


859 


399 


404 


409 


414 


420 


425 


43° 


435 


440 


445 


2 


1 


2 


860 


45° 


455 


460 


465 


470 


475 


480 


485 


490 


495 


3 


1 


8 


86l 


500 


5°5 


5io 


5i5 


520 


526 


53i 


53 6 


54i 


546 


4 
5 


2 

3 


4 



862 


55i 


556 


561 


566 


57i 


576 


58i 


586 


59i 


596 


6 


3 


6 


863 


601 


606 


611 


616 


621 


626 


631 


636 


641 


646 


7 
8 


4 


2 
8 


864 


651 


656 


661 


666 


671 


676 


682 


687 


692 


697 


9 


4 
5 


4 


865 


702 


707 


712 


717 


722 


727 


732 


737 


742 


747 




866 


75 2 


757 


762 


767 


772 


777 


782 


787 


792 


797 




867 


802 


807 


812 


817 


822 


827 


832 


837 


842 


847 




868 


852 


%7 


862 


867 


872 


877 


882 


887 


892 


897 




869 


902 


907 


912 


917 


922 


927 


932 


937 


942 


947 




870 


95 2 


957 


962 


967 


972 


977 


982 


987 


992 


997 




871 


94 002 


007 


012 


017 


022 


027 


032 


°37 


042 


047 




872 


052 


o57 


062 


067 


072 


077 


082 


086 


091 


096 


j 


5 


873 


IOI 


106 


in 


116 


121 


126 


131 


136 


141 


146 


2 


1 






874 


151 


i5 6 


161 


166 


171 


176 


181 


186 


191 


196 


3 

4 
5 


1 


5 


875 


201 


206 


211 


216 


221 


226 


231 


236 


240 


245 


2 


5 


876 


250 


255 


260 


265 


270 


275 


280 


285 


290 


295 


6 


3 





877 


300 


3°5 


310 


3i5 


320 


325 


33° 


335 


34o 


345 


7 
8 


3 
4 


5 



878 


349 


354 


359 


3 6 4 


3 6 9 


374 


379 


384 


389 


394 


9 


4 


5 


879 


399 


404 


409 


414 


419 


424 


429 


433 


438 


443 




880 


448 


453 


458 


463 


468 


473 


478 


483 


488 


493 




881 


498 


5°3 


5°7 


512 


5i7 


522 


527 


532 


537 


542 




882 


547 


552 


557 


562 


567 


57i 


57 6 


58i 


586 


59i 




883 


596 


601 


606 


611 


616 


621 


626 


630 


635 


640 




884 


645 


650 


655 


660 


665 


670 


675 


680 


685 


689 




885 


694 


699 


704 


709 


714 


719 


724 


729 


734 


738 




886 


743 


748 


753 


758 


763 


768 


773 


778 


783 


787 


1 


4 
0.4 


887 


792 


797 


802 


807 


812 


817 


822 


827 


832 


836 


2 





8 


888 


841 


846 


851 


856 


861 


866 


871 


876 


880 


885 


3 

4 


I 
1 


2 
6 


889 


890 


895 


900 


905 


910 


9 J 5 


919 


924 


929 


934 


5 


2 





890 


939 


944 


949 


954 


959 


963 


968 


973 


978 


983 


6 

7 
8 


2 


4 
8 


891 


988 


993 


998 


*002 


*oo7 


*OI2 


*oi7 


*02 2 


*027 


*032 


3 


2 


892 


95 °3 6 


041 


046 


05I 


056 


o6l 


066 


071 


075 


080 


9 


3 


6 


893 


085 


090 


°95 


IOO 


i°5 


IO9 


114 


119 


124 


129 




894 


134 


i39 


i43 


I48 


153 


158 


163 


l68 


173 


177 




895 


182 


187 


192 


197 


202 


207 


211 


2l6 


221 


226 




896 


231 


236 


240 


245 


250 


255 


260 


265 


270 


274 




897 


279 


284 


289 


294 


299 


3°3 


308 


3*3 


318 


323 




898 


328 


33 2 


337 


342 


347 


352 


357 


361 


366 


371 




899 


376 


381 


386 


39° 


395 


400 


405 


410 


415 


419 





III 



LOGARITHMS OF NUMBERS. 



No. 





1 


2 


3 


4 


5 

448 


6 


7 


8 


9 


r 

Pp. Pts. 


900 


95 424 


429 


434 


439 


444 


453 


458 


463 


46S 




901 


472 


477 


482 


487 


49 2 


497 


5°i 


506 


5ii 


5i6 




902 


521 


525 


53° 


535 


540 


545 


55° 


554 


559 


5 6 4 




903 


569 


574 


578 


583 


588 


593 


598 


602 


607 


612 




904 


617 


622 


626 


631 


636 


641 


646 


650 


655 


660 




905 


665 


670 


674 


679 


684 


689 


694 


698 


7°3 


708 




906 


7i3 


718 


722 


727 


732 


737 


742 


746 


75i 


756 




907 


761 


766 


770 


775 


780 


785 


789 


794 


799 


804 




908 


809 


813 


818 


823 


828 


832 


^37 


842 


847 


852 




909 


856 


861 


866 


871 


875 


880 


885 


890 


895 


899 




910 


904 


909 


914 


918 


923 


928 


933 


938 


943 


947 




911 


95 2 


957 


961 


966 


971 


976 


980 


985 


990 


995 




912 


999 


*oo4 


*oo9 


*oi4 


*oi9 


*023 


*028 


*°33 


*o 3 8 


*042 




. 5 

0.5 


913 


96 047 


052 


o57 


061 


0C6 


071 


076 


c£o 


085 


090 


I 


914 


095 


099 


104 


109 


114 


118 


123 


128 


i33 


137 


2 


1 





915 


142 


147 


152 


156 


161 


166 


171 


175 


180 


185 


3 

4 


1 
2 


5 



916 


190 


194 


199 


264 


209 


213 


218 


223 


227 


232 


5 


2 


5 


917 


2 37 


242 


246 


25 1 


256 


261 


265 


270 


275 


280 


6 

7 


3 

3 




5 


918 


284 


289 


294 


298 


3°3 


308 


3i3 


3*7 


322 


327 


8 


4 





919 


33 2 


33 6 


34i 


346 


35° 


355 


360 


365 


3 6 9 


374 


9 


4-5 


920 


379 


384 


388 


393 


398 


402 


407 


412 


417 


421 




921 


426 


43i 


435 


440 


445 


45° 


454 


459 


464 


468 




922 


473 


478 


483 


487 


49 2 


497 


5oi 


506 


5ii 


5i5 




923 


520 


525 


53° 


534 


539 


544 


548 


553 


558 


562 




924 


567 


572 


577 


58i 


586 


59i 


595 


600 


605 


609 




925 


614 


619 


624 


628 


6 33 


638 


642 


647 


652 


656 




926 


661 


666 


670 


6 75 


680 


685 


689 


694 


699 


7°3 




927 


708 


7i3 


717 


722 


727 


73i 


736 


741 


745 


75o 




928 


755 


759 


764 


769 


774 


778 


783 


788 


792 


797 




929 


802 


806 


811 


816 


820 


825 


830 


834 


839 


844 




930 


848 


853 


858 


862 


867 


872 


876 


881 


886 


890 




4 

4 


931 


%5 


900 


904 


9 C 9 


914 


918 


923 


928 


93 2 


937 


1 


932 


942 


946 


95i 


956 


960 


965 


970 


974 


979 


984 


2 





8 


933 


988 


993 


997 


*002 


*oo7 


*OII 


*oi6 


*02I 


*02 5 


*03o 


3 

4 


1 
1 


2 
6 


934 


97 °35 


°39 


044 


049 


°53 


058 


063 


067 


072 


077 


5 


2 





935 


081 


086 


090 


°95 


100 


104 


109 


114 


Il8 


123 


6 

►7 


2 

2 


4 
8 


936 


128 


132 


i37 


142 


146 


151 


i55 


l6o 


165 


169 


# 
8 


3 


2 


937 


174 


179 


183 


188 


192 


197 


202 


206 


211 


216 


9 


3 6 


938 


220 


225 


230 


234 


239 


243 


248 


2 53 


257 


262 




939 


267 


271 


276 


280 


285 


290 


294 


299 


3°4 


308 




940 


3*3 


3*7 


322 


3 2 7 


33* 


33^ 


34o 


345 


35° 


354 




941 


359 


3 6 4 


368 


373 


377 


382 


387 


39i 


396 


400 




942 


405 


410 


414 


419 


424 


428 


433 


437 


442 


447 




943 


45i 


456 


460 


465 


470 


474 


479 


483 


488 


493 




944 


497 


502 


506 


5ii 


5i6 


520 


525 


529 


534 


539 




945 


543 


548 


552 


557 


562 


566 


57i 


575 


580 


585 




946 


589 


594 


598 


603 


607 


612 


617 


621 


626 


630 




947 


635 


640 


644 


649 


653 


658 


663 


667 


672 


676 




948 


681 


685 


690 


695 


699 


704 


708 


7i3 


717 


722 




949 


727 


73i 


736 


740 


745 


749 


754 


759 


7 6 3 


768 





112 



LOGARITHMS OF NUMBERS. 



No. 





I 


2 


3 


4 


5 


6 


7 


8 


9 


Ep. Pts. 


950 


97 772 


777 


782 


786 


791 


795 


800 


804 


809 


813 




951 


818 


823 


827 


832 


836 


841 


845 


850 


855 


859 




952 


864 


868 


873 


877 


882 


886 


891 


896 


900 


9°5 




953 


909 


914 


918 


923 


928 


932 


937 


941 


946 


95° 




954 


955 


959 


964 


968 


973 


978 


982 


9S7 


991 


996 




955 


98 000 


005 


009 


014 


019 


023 


028 


032 


°37 


041 




956 


046 


050 


055 


059 


064 


068 


o73 


078 


082 


087 




957 


091 


096 


IOO 


io 5 


109 


114 


118 


123 


127 


132 




958 


*37 


141 


146 


i5° 


J 55 


159 


164 


168 


t-73 


177 




959 


182 


186 


191 


195 


200 


204 


209 


214 


21S 


223 




960 


227 


232 


236 


241 


245 


250 


254 


259 


263 


268 




961 


272 


277 


281 


286 


290 


295 


299 


3°4 


308 


3i3 




962 


318 


322 


327 


33 1 


33 6 


34o 


345 


349 


354 


358 






963 


363 


367 


372 


376 


3Si 


385 


39° 


394 


399 


403 


1 


5 

O- c 


964 


408 


412 


417 


421 


426 


43° 


435 


439 


444 


448 


2 


1 


.0 


965 


453 


457 


462 


466 


471 


475 


480 


484 


489 


493 


3 

4 


1 
2 


•5 




966 


498 


502 


5°7 


5ii 


5i6 


520 


525 


5 2 9 


534 


538 


5 


2 


5 


967 


543 


547 


552 


556 


56i 


565 


57° 


574 


579 


583 


6 


3 





968 


588 


592 


597 


601 


605 


610 


614 


619 


623 


628 


7 

8 


3 
4 


5 




969 


632 


637 


641 


646 


650 


655 


659 


664 


668 


^73 


9 


4 


5 


970 


677 


682 


686 


691 


695 


700 


704 


709 


7i3 


717 




971 


722 


726 


73i 


735 


740 


744 


749 


753 


758 


762 




972 


767 


771 


776 


780 


784 


789 


793 


798 


802 


807 




973 


811 


816 


820 


825 


829 


834 


838 


843 


847 


851 




974 


856 


860 


865 


869 


874 


.878 


883 


887 


892 


896 




975 


900 


905 


909 


914 


918 


923 


927 


932 


936 


941 




976 


945 


949 


954 


958 


963 


967 


972 


976 


981 


985 




977 


989 


994 


998 


*oo3 


*oc>7 


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978 


99 °34 


038 


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047 


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061 


065 


069 


074 




979 


078 


083 


087 


092 


096 


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114 


118 




980 


123 


127 


131 


136 


140 


145 


149 


154 


158 


162 




981 


167 


171 


176 


180 


185 


189 


i93 


I98 


202 


207 


1 


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982 


211 


216 


220 


224 


229 


2 33 


238 


242 


247 


251 


2 





8 


983 


2 55 


260 


264 


269 


2 73 


277 


282 


286 


291 


295 


3 

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984 


300 


3°4 


308 


3*3 


3i7 


322 


326 


33° 


335 


339 


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985 


344 


348 


352 


357 


361 


366 


37o 


37 Ar 


379 


383 


6 


2 


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8 
2 


986 


388 


392 


396 


401 


405 


410 


414 


419 


423 


427 


7 
8 


2 
3 


987 


43 2 


43 6 


44i 


445 


449 


454 


458 


463 


467 


47i 


9 


3 


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988 


476 


480 


484 


489 


493 


498 


502 


506 


5ii 


5i5 




989 


520 


5^4 


528 


533 


537 


542 


546 


55° 


555 


559 




990 


564 


568 


572 


577 


581 


585 


59o 


594 


599 


603 




991 


607 


612 


616 


621 


625 


629 


634 


638 


642 


647 




992 


651 


656 


660 


664 


669 


673 


677 


682 


686 


691 




993 


695 


699 


704 


708 


712 


7i7 


721 


726 


73° 


734 




994 


739 


743 


747 


75 2 


756 


760 


765 


769 


774 


778 




995 


782 


787 


791 


795 


800 


804 


808 


813 


817 


822 




996 


826 


830 


835 


839 


843 


848 


852 


856 


861 


865 




997 


870 


874 


878 


883 


887 


891 


896 


900 


904 


909 




998 


9i3 


917 


922 


926 


93° 


935 


939 


944 


948 


952 




| 999 


957 


961 


965 


970 


974 


978 


983 


987 


991 


996 





113 



APPENDIX A 



The following notes and tables relating to drill capacities 
and losses due to valves, elbows and tees are taken from the 
Ingersoll-Rand catalog. 

DRILL CAPACITY TABLES 

The following tables are to determine the amount of free 
air required to operate rock drills at various altitudes with 
air at given pressures. 

The tables have been compiled from a review of a wide 
experience and from tests run on drills of various sizes. They 
are intended for fair conditions in ordinary hard rock, but 
owing to varying conditions it is impossible to make any 
guarantee without a full knowledge of existing conditions. 

In soft material where the actual time of drilling is short, 
more drills can be run with a given sized compressor than 
when working in hard material, when the drills would be 
working continuously for a longer period, thereby increasing 
the chance of all the drills operating at the same time. 

In tunnel work, where the rock is hard, it has been the 
experience that more rapid progress has been made when the 
drills were operated under a high air pressure, and that it 
has been found profitable to provide compressor capacity in 
excess of the requirements by about 25 per cent. There is 
also a distinct advantage in having a compressor of large 
capacity, in that it saves the trouble and expense of 
moving the compressor as the work progresses, and will 
not interfere with the progress of the work by crowding the 
tunnel. 

No allowance has been made in the tables for loss due to 

leaky pipes, or for transmission loss due to friction, but the 

capacities given are merely the displacement required, bo 

114 



APPENDIX A 



115 



that when selecting a compressor for the work required these 
matters must be taken into account. 

Table I gives cubic feet of free air required to operate one 
drill of a given size and under a given pressure. 

Table II gives multiplication factors for altitudes and 
number of drills by which the air consumption of one drill 
must be multiplied in order to give the total amount of air. 



TABLE I. — CUBIC FEET OF FREE AIR REQUIRED TO RUN 
ONE DRILL OF THE SIZE AND AT THE PRESSURE 
STATED BELOW 



3 

CO M 

03 t-j 

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to ^ 
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60 
70 
80 
90 
100 


SIZE AND CYLINDER DIAMETER OF DRILL 


A3 5 

2" 


A32 


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68 
77 
86 
95 
104 


C 
2f" 

82 

93 

104 

115 

126 


D 
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90 
102 
114 
126 
138 


D 

95 
108 
120 
133 
146 


D 

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97 
110 
123 
136 
149 


E 
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F 

113 
129 
143 
159 
174 


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130 

147 
164 
182 
199 


H 

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150 
170 

190 
210 
240 


H9 

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164 
181 
207 
230 
252 


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50 
56 
63 

70 

77 


60 

68 
76 
84 
92 


100 
113 
127 
141 
154 


108 
124 
131 
152 
166 



116 



APPENDIX A 



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APPENDIX A 117 



GLOBE VALVES, TEES AND ELBOWS 

The reduction of pressure produced by globe valves is the 
same as that caused by the following additional lengths 
of straight pipe, as calculated by the formula : 

.,,.,. ,, Al , . 114 X diameter of pipe 

Additional length of pipe = — -— r . — —- 

1 + (36 -r- diameter) 

Diameter of pipe ) 1 1J 2 2\ 3 3£ 4 5 6 inches 

Additional length (24 7 10 13 16 20 28 36 feet 

7 8 10 12 15 18 20 22 24 inches 
44 53 70 88 115 143 162 181 200 feet 

The reduction of pressure produced by elbows and tees is 
equal to two-thirds of that caused by globe valves. The 
following are the additional lengths of straight pipe to be 
taken into account for elbows and tees. For globe valves 
multiply by |. 

Diameter of pipe | 1 1| 2 2\ 3 Z\ 4 5 6 inches 
Additional length J 2 3 5 7 9 11 13 19 24 feet 

7 8 10 12 15 18 20 22 24 inches 
30 35 47 59 77 96 108 120 134 feet 

These additional lengths of pipe for globe valves, elbows 
and tees must be added in each case to the actual lengths 
of straight pipe. Thus a 6-inch pipe, 500 feet long, with 
1 globe valve, 2 elbows and 3 tees, would be equivalent to 
a straight pipe 500 + 36 + (2 X 24) + (3 X 24) = 656 feet 
long. 



APPENDIX B 



In the following tables are collected all the reliable data 
that the author has been able to find relative to friction in 
air pipes. 

In these tables the significance of the symbols is as fol- 
lows: 

No = Reference number of the experiment. 

Pi = Absolute pressure at first station on the pipe = 
pounds per square inch. 

2>2 — Absolute pressure at second station on the pipe = 
pounds per square inch. 

Vm — ~T = mean pressure in pipe between stations. 

f = Pi — V2 = pressure lost between stations = pounds 
per square inch. 

r = Mean ratio of compression between stations. 

v a = Cubic feet of free air passing per second. 

v m = Cubic feet of compressed air passing per second. 

s = Velocity of air in pipe = feet per second. 

Q = Weight in pounds of air passing per second. 

d = Diameter of pipe in inches. 

I = Length of pipe in feet. 

c = Coefficient in formula (20), Art. 23, viz., f = c — — • 

a 5 r 



118 



APPENDIX B 



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MP I * 182? 



